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Full text of "Plane trigonometry and numerical computation"

P n ilLD H PLANE TRIGONOMETRY -. AND NUMERICAL COMPUTATION /~\ ^^^^^^ JOHN ALEXANDER JAMESON, Jr. 1903-1934 This book belonged to John Alexander Jameson, Jr., A.B., Wil- liams, 1925; B.S., Massachusetts Institute of Technology, 1928; M.S., California, 1933. He was a member of Phi Beta Kappa, Tau Beta Pi, the American Society of Civil Engineers, and the Sigma Phi Fraternity. His untimely death cut short a promising career. He was engaged, as Research Assistant in Mechanical Engineering, upon the design and construction of the U. S. Tidal Model Labora- tory of the University of California. His genial nature and unostentatious effectiveness were founded on integrity, loyalty, and devotion. These qualities, recognized by everyone, make his life a continuing beneficence. Memory of him will not fail among those who knew him. PLANE TRIGONOMETRY AND NUMERICAL COMPUTATION BY JOHN WESLEY YOUNG »/ PROFESSOR OF MATHEMATICS DARTMOUTH COLLEGE AND FRANK MILLETT MORGAN AS8I8TANT PROFESSOR OF MATHEMATICS DARTMOUTH COLLEGE r°. .ss <+ Mi hi l p *c is l * * ■ ui y If Neta gork * * THE MACMILLAN COMPANY 1919 All rights reserved I C /> « cat 1 l'£&f*' SAS35 Copyright, 1919, Bv THE MACM1LLAN COMPANY. Set up and electrotyped. Published October, 1919. ENGINEERING Uftfttjty. n.,1 p (AiJiXi.i ; ..!P,...' — Norfoootr Pteaa /| 1 * ^- .1. S_ Olshinor O.n 'Ro^T.rJnU JL Q^UU <^_ J. S. Cushing Co. — Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE Ever since the publication of our Elementary Mathematical Analysis (The Macmillan Co., 1917) we have been asked by numerous teachers to publish separately, as a textbook in plane trigonometry, the material on trigonometry and logarithms of the text mentioned. The present textbook is the direct outcome of these requests. Of course, such separate publication of material taken out of the body of another book necessitated some changes and an in- troductory chapter. As a matter of fact, however, we have found it desirable to make a number of changes and additions not required by the necessities of separate publication. As a result fully half of the material has been entirely rewritten, with the purpose of bringing the text abreast of the most recent tendencies in the teaching of trigonometry. There is an increasing demand for a brief text emphasizing the numerical aspect of trigonometry and giving only so much of the theory as is necessary for a thorough understanding of the numerical applications. The material has therefore been ar- ranged in such a way that the first six chapters give the essen- tials of a course in numerical trigonometry and logarithmic computation. The remainder of the theory usually given in the longer courses is contained in the last two chapters. More emphasis than hitherto has been placed on the use of tables. For this purpose a table of squares and square roots has been added. Recent experience has emphasized the appli- cations of trigonometry in navigation. We have accordingly added some material in the text on navigation, have introduced v 889757 vi PREFACE the haversine, and have added a four-place table of haversines for the benefit of those teachers who feel that the use of the haversine in the solution of triangles is desirable. This material can, however, be readily omitted by any teacher who prefers to do so. J. W. Young, F. M. Morgan. Hanover, N.H., August, 1919. CONTENTS CHAPTER PAGES I. Introductory Conceptions 1-10 II. The Right Triangle . . . . . . 11-31 III. Simple Trigonometric Relations . . . . 32-39 IV. Oblique Triangles . 40-49 V. Logarithms 50-60 VI. Logarithmic Computation 61-74 VII. Trigonometric Relations 75-87 VIII. Trigonometric Relations (continued) . . . 88-103 Tables 106-119 Index 121-122 vu PLANE TRIGONOMETRY AND NUMERICAL COMPUTATION CHAPTER I INTRODUCTORY CONCEPTIONS 1. The Uses of Trigonometry. The word " trigonometry " is derived from two Greek words meaning " the measurement of triangles." A triangle has six so-called elements (or parts) • viz., its three sides and its three angles.' AV'e v *rk s w from our study of geometry that, in general, if three elements of a tri- angle (not all angles) are given, the triangle is completely determined.* Hence, if three such determining elements of a triangle are given, it should be possible to compute the remain- ing elements. The methods by which this can be done, i.e. methods for " solving a triangle," constitute one of the prin- cipal objects of the study of- trigonometry. If two of the angles of a triangle are given, the third angle can be found from the relation A + B -f- C = 180° (A, B, and C representing the angles of the triangle) ; also, in a right tri- angle, if two of the sides are known, the third side can be found from the relation a 2 + b 2 = c- (a, b being the legs and c the hypotenuse). But this is nearly the limit to which the methods of elementary geometry will allow us to go in the solution of a triangle. Trigonometry f is the foundation of the art of surveying * What exceptions are there to this statement ? t Throughout this book we shall confine ourselves to the subject of "plane trigonometry," which deals with rectilinear triangles in a plane. " Spherical trigonometry" deals with similar problems regarding triangles on a sphere whose sides are arcs of great circles. B 1 PLANE TRIGONOMETRY H, and of much of the art of navigation. It is, moreover, of primary importance in practically every branch of pure and applied mathematics. Many of the more elementary applica- tions will be presented in later portions of this text. 2. The " Shadow Method." The ancient Greeks employed the theory of similar triangles in the solution of a special type of triangle problem which it is worth our while to examine briefly, because it contains the germ of the theory of trigo- nometry. It is desired to find the height CA of a vertical tower stand- ing on a level plain. It is observed that at a certain time the tower casts a shadow 42 ft. long. At the same time a pole C'A', 10 ft. long, held vertically with one end on the ground casts a shadow 7 ft. long. From these data the height of the tower is readily com- puted as follows : The right triangles ABC and A'B'C are similar since Z B = Z B'. (Why ?) Therefore we have CA = C'A' 10 BC A A: B' 7 C' or CA = B'C C'A' B'C The tower is then 60 ft. high. 3. A " Function " of an Angle. £<7 = y x42 = 60. From the point of view of our future study the important thing to notice in the solution CA C'A' of the preceding article is the fact that the ratios , — — v Bkj b g are equal, i.e. that the ratio of the side opposite the angle B to the side adjacent to the angle is determined by the size of the angle, and does not depend at all on any of the other elements of the triangle, provided only it is a right triangle. I, § 3] INTRODUCTORY CONCEPTIONS 3 Definition. Whenever a quantity depends for its value on a second quantity, the first is called a function of the second. Thus in our example the ratio of the side opposite an angle of a right triangle to the side adjacent is a quantity which depends for its value only on the angle ; it is, therefore, called a function of the angle. This ratio is merely one of several functions of an angle which we shall define in the next chapter. By means of these functions the fundamental prob- lem of trigonometry can be readily solved. The particular function which we have discussed is called the tangent of the angle. Explicitly defined for an acute angle of a right triangle, we have tangent of angle = ^ide op posite the ang le_. side adjacent to the angle If the angle B in the preceding example were measured it would be found to contain 55°. In any right triangle then containing an angle of 55° we should find this ratio to be equal to - T -, or 1.43. If the angle is changed, this ratio is changed, but it is fixed for any given angle. If the angle is 45°, the tangent is equal to 1, since in that case the triangle is isosceles. The word tangent is abbreviated " tan." Thus we have already found tan 55° = 1.43 and tan 45° = 1.00. Similarly to every other acute angle corresponds a definite number, which is the tangent of that angle. The values of the tan- gents of angles have been tabulated. ^Ve shall have occasion to use such tables extensively in the future. \ If a, 6, c are the sides of a right triangle ABC with right angle at C and with the usual notation whereby the side a is opposite the angle A and side b opposite the angle B, the defi- nition of the tangent gives tanjB = -. a PLANE TRIGONOMETRY [I, §3 From this we get at once, b = a tan B and a = tan B These are our first trigonometric formulas. By means of them and a table of tangents we can compute either leg of a right triangle, if the other leg and an acute angle are given. EXERCISES 1. What is meant by "the elements of a triangle " ? by " solving a triangle ' ' ? 2. A tree casts a shadow 20 ft. long, when a vertical yardstick with one end on the ground casts a shadow of 2 ft. How high is the tree ? 3. A chimney is known to be 90 ft. high. How long is its shadow when a 9-foot pole held vertically with one end on the ground casts a shadow 5 ft. long ? 4. Give examples from your own experience of quantities which are functions of other quantities. 5. Define the tangent of an acute angle of a right triangle. Why does its value depend only on the size of the angle ? 6. In the adjacent figure think of the line BA as rotating about the point B in the direction of the arrow, starting from the position BC (when the angle B is 0) and assum- ing successively the positions BA h BA%, BA 3 , Show that the tangent of the angle B is very small when B is very small, that tan B increases as the angle increases, that tan B is less than 1 as long as B is less than 45°, that tan 45° = 1, that tan B is greater than 1 if the angle is greater than 45°, and that tan B increases without limit as B ap- proaches 90°. 7. The following table gives the values of the tan- gent for certain values of the angle i angle 10° 20° 30° 40° 50° 60° 70° tangent 0.176 0.364 0.577 0.839 1.19 1.73 2.75 I, §4] INTRODUCTORY CONCEPTIONS // (9) 60° 20° By means of this table find the other leg of a right triangle ABC from the elements given : ^ (a) B = 50°, a = 10 (d) B = 20°, b = 13 (6) B = 70°, a = 16 (e) A = 30°, 6=5 (c) B = 40°, & = 24 (/) A = 10°, & = 62 8. From the data and the results of the preceding exercise find the other acute angle and the hypotenuse of each of the right triangles. ^4. Coordinates in a Plane. The student should already be familiar from his study of algebra with the method of locating points in a plane by means of coordinates. Since we shall often have occasion to use such a method in the future, we will recall it briefly at this point. The method consists in referring the points in question to two straight lines X'X and Y l Y, at right angles to each other, which are called the axes of Coordinates. X'X is USUally Second Quadrant drawn horizontally and is called the x-axis ; Y' Y, which is then vertical, is called the y-axis. The position of any point P is completely determined if its distance (measured in terms of some convenient unit) and its direction from each of the axes is known. Thus the position of P x (Fig. 2) is known, if we know that it is 4 units to the right of the ?/-axis and 2 units above the x-axis. If we agree to consider distance measured to the right or upwards as positive, and therefore distance measured to the left or down- ward as negative ; and if, furthermore, we represent distances and directions measured parallel to the x-axis by x, and distances and directions measured parallel to the y-axis by y, then the position of P x may be completely given by the specifications » = -r-4, 2/=-|-2; or more briefly still by the symbol (4, 2). M, X' M s O Third Quadrant Mt Mt rl Fourth Quadrant Fig. 2 6 PLANE TRIGONOMETRY [I, § 4 Similarly, the point P 2 in Fig. 2 is completely determined by the symbol (-3, 5). Observe that in such a symbol the x of the point is written first, the y second. The two numbers x and y, determining the position of a point, are called the coordinates of the point, the x being called the x-coordinate or abscissa, the y being called the y-coordinate or ordinate of the point. What are the coordinates of P 3 and P A in Fig. 2? The two axes of coordinates divide the plane into four regions called quadrants, numbered as in Fig. 2. The quadrant in which a point lies is completely determined by the signs of its coordinates. Thus points in the first quadrant are character- ized by coordinates (+, -+-), those in the second by ( — , +), those in the third by ( — , — ), and those in the fourth by (-f, — ). Square-ruled paper (so-called coordinate or cross section paper) is used to advantage in " plotting " (i.e. locating) points by means of their coordinates. 5. Magnitude and Directed Quantities. In the last article we introduced the use of positive and negative numbers, i.e. the so-called signed numbers, while in the preceding articles, where we were concerned with the sides and angles of triangles, we dealt only with unsigned numbers. The latter represent magnitude or size only (as a length of 20 ft.), while the former represent both a magnitude and one of two opposite direc- tions or senses (as a distance of 20 ft. to the left of a given line). We are thus led to consider two kinds of quantities : (1) magnitudes, and (2) directed quantities. Examples of the former are : the length of the side of a triangle, the weight of a barrel of flour, the duration of a period of time, etc. Ex- amples of the latter are : the coordinates of a point, the tem- perature (a certain number of degrees above or below zero), the time at which a certain event occurred (a certain number of hours before or after a given instant), etc. I, § 6] INTRODUCTORY CONCEPTIONS 7 Geometrically, the distinction between directed quantities and mere magnitudes corresponds to the fact that, on the one hand, we may think of the line segment AB as drawn from A to B or from B to A ; and, on the other hand, we may choose to consider only the length of ' ' *~~* ' ' such a segment, irrespective of its direction. Figure 3 exhibits the geometric representation of 5, + 5, and — 5. A segment whose direc- tion is definitely taken account of is called ^'directed segment. The magnitude of a directed quantity is called its absolute value. Thus the absolute value of — 5 (and also of + 5) is 5. Observe that the segments OM u M X P X (Fig. 2) representing the coordinates of P x are directed segments. 6. Directed and General Angles. In elementary geometry an angle is usually defined as the figure formed by two half- lines issuing from a point. However, it is often more serviceable to think of an angle as being generated by the rotation in a plane of a half-line OP about the point as a pivot, start- ing from the initial position OA and ending at the terminal position OB (Fig. 4). We then say that the line OP has generated the angle AOB. Similarly, if OP rotates from the initial position OB, to the terminal position OA, then the angle BOA is said to be generated. Considerations similar to those regarding directed line segments (§ 5) lead us to regard one of the above directions of rotation as positive and x the other as negative. It is of course quite immaterial which one of the two rotations we regard as positive, but we shall assume, from now on, that counterclockwise rotation is positive and clockwise rotation is negative. Still another extension of the notion Fig. 5 8 PLANE TRIGONOMETRY [I, §6 of angle is desirable. In elementary geometry no angle greater than 360° is considered and seldom one greater than 180°. But from the definition of an angle just given, we see that the revolving line OP may make any number of complete revolu- tions before coming to rest, and thus the angle generated may be of any magnitude. Angles generated in this way abound in practice and are known as angles of rotation * When the rotation generating an angle is to be indicated, it is customary to mark the angle by means of an arrow starting at the initial line and ending at the terminal line. Unless some such device is used, confusion is liable to result. In Fig. G 30° 390' 750 1110 Fig. (5 angles of 30°, 390°, 750°, 1110°, are drawn. If the angles were not marked one might take them all to be angles of 30°. 7. Measurement of Angles. For the present, angles will be measured as in geometry, the degree (°) being the unit of measure. A complete revolution is 360°. The other units in this system are the minute ('), of which 60 make a degree, and the second ("), of which 60 make a minute. This system of units is of great antiquity, having been used by the Babylonians. The considerations of the previous article then make it clear that any real number, positive or negative, may represent an angle, the absolute value of the number representing the magnitude of the angle, the sign representing the direction of rotation. v Fig. 7 Consider the angle XOP = 0, whose vertex O coincides with the origin of a system of rectangular coordinates, and whose initial line OX coin- *For example, the minute hand of a clock describes an angle of —180° n 30 minutes, an angle of — 540° in 90 minutes, and an angle of — 720° in 120 ninutes. I, § 8] INTRODUCTORY CONCEPTIONS 9 cides with the positive half of the a;-axis (Fig. 7) . The angle is then said to be in the first, second, third, or fourth quadrant, according as its terminal .line OP is in the first, second, third, or fourth quadrant. 8. Addition and Subtraction of Directed Angles. The meaning to be attached to the sum of two directed angles is analogous to that for the sum of two directed line segments. Let a and b be /* /& two half-lines issuing from the / £, Y same point O and let (ab) repre- sent an angle obtained by rotat- j£F SJ — ' 5~ q ing a half -line from the position jr IG# y a to the position b. Then if we have two angles (a&) and (6c) with the same vertex O, the sum (a6) + (6c) of the angles is the angle represented by the rotation of a half -line from the position a to the position b and then rotating from the position b to the position c. But these two rotations are together equivalent to a single rota- tion from a to c, no matter what the relative positions of a, 6, c may have been. Hence, we have for any three half -lines a, b, c issuing from a point 0, (1) (ab) + (bc)=(ac), (ob) + (bc)=0, (ab) = (cb)-(ca). It must be noted, however, that the equality sign here means " equal, except possibly for multiples of 360V The proof of the last relation is > left as an exercise. ^^ EXERCISES \^\) 1. On square-ruled paper draw two axes of reference and then plot the following points: (2, 3), (- 4, 2), (- 7, - 1), (0, - 3), (2, - 5), (5, 0). 2. What are the coordinates of the origin ? 3. Where are all the points for which x — 2? x =— 3 ? y — — 1 ? y = ±? x = 0? 4. Show that any point P on the 2/-axis has coordinates of the form (0, y) . What is the form of the coordinates of any point on the x-axis ? 5. A right triangle has the vertex of one acute angle at the origin and one leg along the se-axis. The vertex of the other acute angle is at (7, 10). What is the tangent of the angle at O ? *? -\ 6. What angle does the minute hand of a clock describe in 2 hours and 30 minutes ? in 4 hours and 20 minutes ? ' \ / a 7. Suppose that the dial of a clock is transparent so that it may be read from both sides. Two persons stationed at opposite sides of the dial observe the motion of the minute hand. In what respect will the angles described by the minute hand as seen by the two persons differ? 10 PLANE TRIGONOMETRY [I, § 8 i X / 4 8. In what quadrants are the following angles : 87° ? 135° ? — 325° ? 540°? 1500°? -270°? 9. In what quadrant is 0/2 if is a positive angle less than 360° and in the second quadrant ? third quadrant ? fourth quadrant ? 10. By means of a protractor construct 27° + 85° + (— 30°) + 20° + (-45°). 11. By means of a protractor construct — 130° + 56° — 24°. I J CHAPTER II THE RIGHT TRIANGLE 9. Introduction. At the beginning of the preceding chap- ter we described the fundamental problem of trigonometry to be the " solution of the triangle," i.e. the problem of com- puting the unknown elements of a triangle when three of the elements (not all angles) are given. This problem can be solved by finding relations between the sides and angles of a triangle by means of which it is possible to express the un- known elements in terms of the known elements. In order to establish such relations, it has been found desirable to define certain functions of an angle. One such function — the tangent — was introduced in § 3 by way of preliminary illus- tration. In the present chapter, we shall give a new definition of the tangent of an angle and also define two other equally impor- tant functions — the sine and the cosine. It should be noted that the definition given for the tangent in § 3 applies only to an acute angle of a right triangle. For the purposes of a sys- tematic study of trigonometry we require a more general defini- tion, which will apply to any angle, positive or negative, and of any magnitude. Such definitions are given in the next article, in which the notion of a system of coordinates plays a fundamental role, the notion of a triangle not being introduced at all. After considering some of the consequences of our definitions in §§ 11-13, we consider the way in which these definitions enable us to express relations between the sides and angles of a right triangle. These results are then imme- diately applied to the solution of numerical problems by means r of tables and to applications in surveying and navigation. 11 12 PLANE TRIGONOMETRY [II, § 10 10. The Sine, Cosine, and Tangent of an Angle. We may now define three of the functions referred to in § 3. To this end let = XOP (Fig. 9) be any directed angle, and let zyL us establish a system of rectangular coordinates in the plane of the angle such that the initial side OX of the angle is the positive half of the sc-axis, the vertex being at the origin and the y-axis being in the usual position with respect to the #-axis. Let the units on the two axes be equal. Finally, let P be any point other than on the terminal side of the angle 6, and let its coordinates be (x, y). The directed segment OP = r is called the distance of P and is always chosen posi- tive. The coordinates x and y are positive or negative accord- ing to the conventions previously adopted. We then define The sine of 8 = The cosine of 6 = ordinate of P _ y distance of P ~ r abscissa of P x distance of P ™* , * /v ordinate of P y . . _ The tangent of 8 = -r — -. j-p=~, provided x =£ 0.* These functions are usually written in the abbreviated forms sin 0, cos 0, tan 0, respectively ; but they are read as " sine 0" " cosine 0," " tangent 0." It is very important to notice that the values of these functions are independent of the position of the point P on the terminal line. For let P' (x\ y') be any other point on this line. Then from the similar right triangles xyrf and x'y'r 1 it follows that the ratio of any two sides of the triangle xyr is equal in magnitude and sign to the * Prove that x and y cannot be zero simultaneously. t Triangle xyz means the triangle whose sides are x, y, z. II, § 11] THE RIGHT TRIANGLE 13 ratio of the corresponding sides of the triangle x'y'r'. There- fore the values of the functions just defined depend merely on the angle 9. They are one-valued functions of 6 and are called trigonometric functions. Since the values of these functions are defined as the ratios of two directed segments, they are abstract numbers. They may be either positive, negative, or zero. Remembering that r is always positive, we may readily verify that the signs of the three functions are given by the following table. Quadrant Sine Cosine Tangent 1 • + + 2 3 + 4 + 11. Values of the Functions for 45°, 135°, 225°, 315°. In each of these cases the triangle xyr is isosceles. Why? Since the trigonometric functions are independent of the position of the point P on the terminal line, we may choose the legs of the right triangle xyr to be of length unity, which M. &i C^-L '^%\- Fig. 10 gives the distance OP as V2. Figure 10 shows the four angles with all lengths and directions marked. Therefore, 1 sin 45°= ---, V2 cos 45° = sin 135° = — , V2 cos 135° = sin 225° = — , V2 cos 225° = sin 315° = -—, cos 315° = V2 1 i i V2 tan 45° = 1, tan 135° = -1, tan 225° = 1, tan 315° = - 1. 14 PLANE TRIGONOMETRY [II, § 12 12. Values of the Functions for 30°, 150°, 210°, 330°. From geometry we know that if one angle of a right triangle con- tains 30°, then the hypotenuse is double the shorter leg, which is opposite the 30° angle. Hence if we choose the shorter leg (ordinate) as 1, the hypotenuse (distance) is 2, Ml 'I<s^L vz •vT dLL± t» Fig. 11 and the other leg (abscissa) is V3. Figure 11 shows angles of 30°, 150°, 210°, 330° with all lengths and directions marked. Hence we have cos 30°=-^, tan 30* = — , 2 ' V3 sin W-;|, sin 150° = ^, sin 210° = 2' sin 330° = - -, 2' cos 150° = - ^?, tan 150° = - — , 2 V3 cos 210° = V3 2 ' cos 330 c V3 2 : tan 210° = V3 J tan 330° = - V3 13. Values of the Functions for 60°, 120°, 240°, 300°. It is left as an exercise to construct these angles and to prove that sin 60° = ^5, cos 60 c sin 120° = -^, cos 120° = --, 2' sin 240°=-^?, . 2 ' cos 240° = --, 2' sin 300° = -^, 2 cos 300° =1, tan 60°=V3, tanl20° = -V3, tan240°=V3, tan 300° = - V3. II, § 14] THE RIGHT TRIANGLE 15 14. Sides and Angles of a Right Triacgle. Evidently any right triangle ABC can be so placed in a system of coordi- nates that the vertex of either acute angle coincides with the origin O and that the ad'jacent leg lies along the positive end OX of the aj-axis (Fig. 12). The following relations then follow at once from the defini- tions of the sine, cosine, and tangent of § 10. In any right triangle, the trigonometric functions of either acute angle are given by the ratios : the sine the cosine = side opposite the angle hypotenuse side adjacent to the angle the tangent hypotenuse side opposite the angle side adjacent to the angle ' These relations are fundamental in all that follows. They should be firmly fixed in mind in such a way that they can be readily applied to any right triangle in what- ever position it may happen to be (for example as in Fig. 13). The student should be able to reproduce any of the following relations with- out hesitation whenever called for. They should not be memorized, but should be read from an actual or imagined figure : b Fig. 13 sin^l cos A sin B cos B=-, c tan A = - , tan B = Also the known relation : C 2 = a 2 + b 2 . 16 PLANE TRIGONOMETRY [II, § 14 If any two elements (other than the right angle) of a right triangle are given, we can then find a relation connecting these two elements with any unknown element, from which relation the unknown element can be computed. 15. Applications. The angle which a line from the eye to an object makes with a horizontal line in the same vertical plane is called an angle of elevation or an angle of depression, Horizontal Fig. 14 according as the object is above or below the eye of. the ob- server (Fig. 14). Such angles occur in many examples. Example 1. A man wishing to know the distance between two points A and B on opposite sides of a pond locates a point C on the land (Fig. 15) such that AC = 200 rd., angle C = 30°, and angle B = 90°. Find the distance AB. AB AG AB = AC sin G = 200 • sin 30° 100 rd. Solution : sin C. (Why ?) = 200 • * Fig. 15 Example 2. Two men stationed at points A and G 800 yd. apart and in the same vertical plane with a balloon B, observe simultaneously the angles of elevation of the balloon to be 30° and 45° respectively. Find the height of the balloon. Solution : Denote the height of the balloon DB by y, and let DC = x; then AD = 800 - x. L 800-x D x Fig. 16 II, § 15J THE RIGHT TRIANGLE 17 Since tan 45° = 1, we have 1 =-, x 1 y and since tan 30° =s 1/V3, we have — - == — g ^ _ x ' Therefore x = y and 800 — x — y V3. 800 Solving these equations for y, we have y — = 292.8 yd. V3 + 1 EXERCISES • 1. In what quadrants is the sine positive ? cosine negative ? tangent positive ? cosine positive? tangent negative ? sine negative ? 2. In what quadrant does an angle lie if (a) its sine is positive and its cosine is negative ? (6) its tangent is negative and its cosine is positive? (c) its sine is negative and its cosine is positive ? (d) its cosine is positive and its tangent is positive ? 3. Which of the following is the greater and why : sin 49° or cos 49° ? £in 35° or cos 35° ? 4. If 6 is situated between 0° and 360°, how many degrees are there in 6 if tan = 1? Answer the similar question for sin = % ; tan $ = — 1 . 5. Does sin 60° = 2 • sin 30° ? Does tan 60° = 2 • tan 30° ? What can you say about the truth of the equality sin 2 = 2 sin 6 ? M) The Washington Monument is 555 ft. high. At a certain place in the plane of its base, the angle of elevation of the top is 60°. How far is that place from the foot and from the top of the tower ? — "^. A boy whose eyes are 5 ft. from the ground stands 200 ft. from a flagstaff. From his eyes, the angle of elevation of the top is 30°. How high is the flagstaff ? 8. A tree 38 ft. high casts a shadow 38 ft. long. What is the angle of elevation of/the top of the tree as seen from the end of the shadow ? How far is i*4rom the end of the shadow to the top of the tree ? i'rom the top of a tower 100 ft. high, the angle of depression of two stones, which are in a direction due east and in the plane of the base are 45° and 30° respectively. How far apart are the stones ? .4ns. 100( V3 - 1) = 73.2 ft. 18 PLANE TRIGONOMETRY [II, § 15 10. Find the area of the isosceles triangle in which the equal sides 10 inches in length include an angle of 120°. Ans. 25 V3 = 43.3 sq. in. -^11. Is the formula sin 2 = 2 sin cos true when = 30° ? 60° ? 120°? <l2! From a figure prove that sin 117° = cos 27°. 13. Determine whether each of the following formulas is true when = 30°, 60°, IHS , 210 D : 1 + tan 2 = — - — COS 2 ' 1 + - 1 - — *-, tan 2 sin 2 sin 2 -f cos 2 i 1. ,""i4. Let Pi(Xi, ?/i) and P-z(x2, yt) be any two points the distance be- tween which is r (the units on the axes being equal) . If is the angle that the line PiP% makes with the x-axis, prove that x 2 - Xi , ?/2 *r=^» = 2 r. }l6. Computation of the Value of One Trigonometric Function from that of Another. J>±£Si Fig. 17 Example 1. Given that sin = f, find the values of the other functions. Since sin is positive, it follows that is an angle in the first or in the second quad- rant. Moreover, since the value of the sine is |, then y = 3 • k and r = 5 • k, where k is any positive constant different from zero. (Why?) It is, of course, immaterial what positive value we assign to k, so we shall assign the value 1. We know, however, that the abscissa, ordinate, and distance are connected by the relation x 2 + y 2 = r 2 , and hence it follows that x = ± 4. Figure 17 is then self-explanatory. Hence we have, for the first quadrant, sin = f , cos = f , and tan = £ ; for the second quadrant, sin = |, cos = — |, tan = — f . is negative, find the other trigonometric functions of the angle 0. Since sin is positive and tan is negative, must be in the second quadrant. We can, therefore, con- struct the angle (Fig. 18), and we obtain sin = ^ T , cos = — Y§, tan = — T \. Fig. 18 II, § 17] THE RIGHT TRIANGLE 19 k 17. Computation for Any Angle. Tables. The values of the trigonometric functions of any angle may be computed by the graphic method. For example, let us find the trigonometric functions of 35°. We first construct on square-ruled paper, by means of a protractor, an angle of 35° and choose a point P on the ter- minal line so that OP shall equal 100 units. Then from the figure we find that 0^=82 units and MP = 57 units. Therefore TOT _L'. -; :: -l >0~ TPT ■':':': -[jjT -■■:■;■■ V* :: : rj:::i b ::: .ft- ■•-■ ■to — : — ■i'>:.\.'..;. A : ' • :: :i\ m V * 10 XV SO 40 SO 60 70 (SO 90 100 Fig. 19 sin 35° = tVv = °- 57 > cos 35 ° = Tiro = °- 82 > tan 35 ° = U = 0.70. The tangent may be found more readily if we start by tak- ing OA = 100 units and then measure AB. In this case, AB = 70 units and hence tan3o° = ^^ = 0.70. It is at once evident that the graphic method, although simple, gives only an approximate result. However, the values of these functions have been computed accurately by methods beyond the scope of this book. The results have been put in tabular form and are known as tables of natural trigonometric functions. Such tables and how to use them will be discussed in the next article. Figure 20 makes it possible to read off the sine, cosine, or tangent of any angle between 0° and 90° with a fair degree of accuracy. The figure is self-explanatory. In reading off values of the tangent use the vertical line through 100 for angles up to 55°, and the line through 10 for angles greater than 55°. Its use is illustrated in some of the following exercises. 20 PLANE TRIGONOMETRY [II, § 17 10 Fig so to 20. — Graphical, T 60 60 70 60 90 100 able oe Trigonometric Functions II, § 18] THE RIGHT TRIANGLE 21 EXERCISES Find the other trigonometric functions of the angle 6 when t£)tan0 = -3. 3. cos = 1$. 5. sin0 = f. 2. sin0 = -|. 4. tan0=f 6. cos0= — |. rl) sin = f and cos is negative. 8. tan = 2 and sin is negative. 9. sin = — \ and tan is positive. 10. cos = § and tan is negative. 11. Can 0.6 and 0.8 be the sine and cosine, respectively, of one and the same angle ? Can 0.5 and 0.9 ? Ans. Yes ; no. 12. Is there an angle whose sine is 2 ? Explain. 13. Determine graphically the functions of 20°, 38°, 70°, 110°. 14. From Fig. 20, find values of the following : sin 10°, cos 50°, tan 40°, sin 80°, tan 70°, cos 32°, tan 14°, sin 14°. 15. A tower stands on the shore of a river 200 ft. wide. The angle of elevation of the top of the tower from the point on the other shore exactly opposite to the tower is such that its sine is \. Find the height of the tower. 16. From a ship's masthead 160 feet above the water the angle of de- pression of a boat is such that the tangent of this angle is / 2 . Find the distance from the boat to the ship. Ans. 640 yards. 18. Use of Tables of Trigonometric Functions. Examina- tion of the tables of " Four Place Trigonometric Functions " (p. 112) shows columns headed " Degrees," " Sine," " Tangent," " Cosine," and under each of the last three named a column headed " Value " (none of the other columns eoncern us at pres- ent). Two problems regarding the use of these tables now present themselves. 1. To find the value of a function when the angle is given. (a) Find the value of sin 15° 20'. In the column headed " Degrees " locate the line corresponding to 15° 20' (p. 113) ; on the same line in the " value " column for the " Sine," we read the result : sin 15° 20' = 0.2644. On the same line, by using the proper column, we find tan 15° 20' = 0.2742, and cos 15° 20' = 0.9644. / H 22 PLANE TRIGONOMETRY [II, § IS (b) Find the value of tan 57° 50'. The entries in the column marked " Degrees " at the top only go as far as 45° (p. 116). But the columns marked " Degrees " at the bottom contain entries beginning with 45° (p. 116) and running back- wards to 90° (p. 112). In using these entries we must use the designations at the bottom of the columns. Thus on the line corresponding to 57° 50' (p. 115) we find the desired value : tan 57° 50' = 1.5900. Also sin 57° 50' = 0.8465, and cos 57° 50' = 0.5324. (c) Find the value of sin 34° 13'. This value lies between the values of sin 34° 10' and 34° 20'. We find for the latter sin 34° 10' = 0.5616 sin 34° 20' = 0.5640 Difference for 10' = 0.0024 Assuming that the change in the value of the function throughout this small interval is proportional to the change in the value of the angle, we conclude that the change for 1' in the angle would be 0.00024. For 3', the change in the value of the function would then be 0.00072. Neglecting the 2 in the last place (since we only use four places and the 2 is less than 5), we find sin 34° 13' = 0.5616 + 0.0007 = 0.5623. This process is called interpolation. With a little practice all the work in- volved can and should be done mentally ; i.e. after locating the place in the table (and marking it with a finger), we observe that the " tabular difference " is " 24 " ; we calculate mentally that .3 of 24 is 7.2, and then add 7 to 5616 as we write down the desired value 0.5623. Similarly we find tan 34° 13' = 0.6800 (the correction to be added is in this case 12.9 which is " rounded off " to 13) and cos 34° 13' = 0.8269. (Observe that in this case the correction must be subtracted. Why ?) 2. To find the angle when a value of a function is given. ere we proceed in the opposite direction. Given sin A = ■J N II, § 18] THE RIGHT TRIANGLE 23 0.3289 ; find A. An examination of the sine column shows that the given value lies between sin 19° 10' ( = 0.3283) and. > sin 19° 20'(= 0.3311). We note the tabular difference to be 28. The correction to be applied to 19° 10' is then fa of 10' = f f ' = - 1 /' = 2.1'. Hence A = 19° 12.1'. (With a four place table do not carry your interpolation farther than the nearest tenth of a minute.) (See § 20.) \ EXERCISES * 1. For practice in the use of tables, verify the following : (a) sin 18° 20' = 0.3145 (d) sin 27° 14' =0.4576 (g) sin 62° 24M =0.8862 (6) cos 37° 30' =0.7934 (e) cos 34° 11' =0.8272 (h) cos 59° 46' .2 =0.5034 (c) tan 75° 50' =3.9617 (/) tan 68° 21' = 2.5173 (i) tan 14° 55'.6 =0.2665 Assume first that the angles are given and verify the values of the functions. Then assume the values of the functions to be given and verify the angles. 2. A certain railroad rises 6 inches for every 10 feet of track. What angle does the track make with the horizontal ? NJ 3. On opposite shores of a lake are two flagstaffs A and B. Per- pendicular to the line AB and along one shore, a line BC = 1200 ft. is measured. The angle ACB is observed to be 40° 20'. Find the distance between the two flagstaffs. 4. The angle of ascent of a road is 8°. If a man walks a mile up the road, how many feet has he risen ? \ 5. How far from the foot of a tower 150 feet high must an observer, 6 ft. high, stand so that the angle of elevation of its top may be 23°. 5 ? 6. From the top of a tower the angle of depression of a stone in the lane of the. base is 40° 20'. What is the angle of depression of the stone from a point halfway down the tower? 7. The altitude of an isosceles triangle is 24 feet and each of the equal angles contains 40° 20'. Find the lengths of the sides and area of the triangle. 8. A flagstaff 21 feet high stands on the top of a cliff. From a point on the level with the base of the cliff, the angles of elevation of the top and bottom of the flagstaff are observed. Denoting these angles by « and /3 respectively, find the height of the cliff in case sin a = -/ 7 and Ans. 75 feet. \ 24 PLANE TRIGONOMETRY [II, § 18 9. A man wishes to find the height of a tower CB which stands on a horizontal plane. From a point A on this plane he finds the angle of ele- vation of the top to be such that sin CAB = f . From a point A' which is on the line AC and 100 feet nearer the tower, he finds the angle of elevation of the top to be such that tan CA'B'= §. Find the height of fche tower. 10. Find the radius of the inscribed and circumscribed circle of a regu- ar pentagon whose side is 14 feet. 11. If a chord of a circle is two thirds of the radius, how large an angle at the center does the chord subtend ? 19. Computation with Approximate Data. Significant Figures. The numerical applications of trigonometry (in sur- veying, navigation, engineering, etc.) are concerned with com- puting the values of certain unknown quantities (distances, angles, etc.) from known data which are secured by measure- ment. Now, any direct measurement is necessarily an approxi- mation. A measurement may be made with greater or less accuracy according to the needs of the problem in hand — but it can never be absolutely exact. Thus, the information on a signpost that a certain village is 6 miles distant merely means that the distance is 6 miles to the nearest mile — i.e. that the distance is between 5± and 6^ miles. Measurements in a physical or engineering laboratory need sometimes to be made to the nearest one ' thousandth of an inch. For example the bore of an engine cylinder may be measured to be 3.496 in., which means that the bore is between 3.4955 in. and 3.4965 in. A simple convention makes it possible to recognize at a glance the degree of accuracy implied by a number represent- ing an approximate measure (either direct or computed). This convention consists simply in the agreement to write no more figures than the accuracy warrants. Thus in arithmetic 6 and 6.0 and 6.00 all mean the same thing. This is not so, when these numbers are used to express the result of measurement or the result of computation from approximate data. Thus 6 means that the result is accurate to the nearest unit, 6.0 that II, § 20] THE RIGHT TRIANGLE 25 it is accurate to the nearest tenth of a unit, 6.00 to the nearest hundredth of a unit. These considerations have an important bearing on practical computation. If the side of a square is measured and found to he 3.6 in. and the length of the diagonal is computed by the formula : diagonal =/ side x V2^4t would be wrong to write = 3.6 x V2 = 3.6 x 1.4142 = 5.09112 in. The correct result is 5.1 in. For the computed value of the diagonal cannot be more accurate than the measured value of the side. The result 5.09112 must therefore be " rounded off " to two significant figures, which gives 5.1. As a matter of fact for the purpose of this problem V2 = 1.4142 should be rounded before multi- plication to V2 = 1.4 ; thereby reducing the amount of labor necessary. A number is " rounded off," by dropping one or more digits at the right and, if the last digit dropped is 5 + , 6, 7, 8, or 9 increasing the preceding digit by 1.* Thus the successive approximations to w obtained by rounding of 3.14159 ••• are 3.1416, 3.142, 3.14, 3.1, 3. 20. The Number of Significant Figures of a number (in the decimal notation) may now be defined as the total number of digits in the number, except that if the number has no digits to the right of the decimal point, any zeros occurring between the decimal point and the first digit different from zero are not counted as significant. Thus, 34.06 and 3,406,000 are both numbers of four significant figures : while 3,406,000.0 is a number of eight significant figures.! * In rounding off a 5 computers round off to an even digit. Thus 1.415 would be rounded to 1.42, whereas 1.445 would be rounded to 1.44. If this rule is used consistently the errors made will tend to compensate each other. t Confusion will arise in only one case. For example, if 3999.7 were rounded by dropping the 7 we should write it as 4000 which according to the above definition would have only 1 significant figure, whereas we know from the way it was obtained that all four figures are significant. In such a case we may underscore the zeros to indicate they are significant or use some other device. 26 PLANE TRIGONOMETRY [II, § 20 In any computation involving multiplication or division the number of significant figures is generally used as a measure of the accuracy of the data. A computed result should not in general contain more significant figures than the least accurate of the data. But computers generally retain one additional figure during the computation and then properly round off the final result. Even then the last digit may be inaccurate — but that is unavoidable. The following general rules will be of use in determining the degree of accuracy to be expected and in avoiding useless labor : 1. Distances expressed to two significant figures call for angles expressed to the nearest 30' and vice versa. 2. Distances expressed to three significant figures call for angles expressed to the nearest 5', and vice versa. 3. Distances expressed to four significant figures call for angles expressed to the nearest minute, and vice versa. 4. Distances expressed to five significant figures call for angles expressed to the nearest tenth of a minute, and vice, versa. In working numerical problems the student should use every safeguard to avoid errors. Neatness and systematic arrange- ment of the work are important in this connection. All work should be checked in one or more of the following ways. 1. Gross errors may be detected by habitually asking oneself : Is this result reasonable or sensible ? 2. A figure drawn to scale makes it possible to measure the unknown parts and to compare the results of such measurements with the computed results. 3. An accurate check can often be secured with com- paratively little additional labor by computing one of the quantities from two different formulas or by verifying a known relation. For example, if the legs a, b of a right tri- angle have been computed by the formulas a = c sin A and b = c cos A, we may check by verifying the relation a 2 + b 2 = c 2 . II, § 21] THE RIGHT TRIANGLE 27 Example. A straight road is to be built from a point A to a point B which is 5.92 miles east and 8.27 miles north of A. What will be the direction of the road and its length ? 5.92 , D 8.27 Formulas : Therefore tan A = AB = 8 27 cos A tan A = 0.716 and A = 35° 35', cos ^ = 0.813 ^£ = 10.17.* Check by a 2 + & 2 = c 2 . From a table of squares (p. 107, see § 21) (5.92) 2 = 35.05 (8.27) 2 = 68.39 (10.17) 2 = 103.4. 103.4 21. Use of Table of Squares. Square Roots. The table of squares of numbers (p. 106) may be used to facilitate com- putation. In the example of the last article, we required the square of 5.92. We find 5.9 on p. 107 in the left-hand column and find the third digit 2 at the head of a certain column. At the intersection of the line and column thus determined we find the desired result (5.92) 2 = 35.05. The square of 8.27 is found similarly at the intersection of the line corresponding to 8.2 and the column headed 7. To find (10.17) 2 , we find the line corresponding to 1.0 (the first two digits, neglecting the decimal point) and find (1.01)* = 1.020 and (1.02) 2 = 1.040. By interpolating, as explained in § 18, we find (1.017) 2 = 1.034. Now shifting the decimal point one place in the "number" requires a corresponding shift of two places in the square. Hence, (10.17)* = 103.4. The table can also be used to find the square root of a num- ber. Thus to find V2 we find, on working backwards in this table, that 2 lies between 1.988 [=(1.41)*] and 2.016 [=(1.42)*]. By interpolation we then find V2 m 1.414, correct to four significant places. [Tabular difference = 28 ; correction = -*^ = 4 in the fourth place.] ♦The retention of four significant figures in AB is justified because the number is so small at the left. ^ 28 PLANE TRIGONOMETRY [II, § 21 EXERCISES 1. From an observing station 357 ft. above the water, the angle of depression of a ship is 2° 15 f . Find the horizontal distance to the ship in yards . 2l A projectile falls in a straight line making an angle of 25° with the horizontal. Will it strike the top of a tree 24 meters high which is 72 meters from the point where the projectile would strike the ground ? 3. At a point 372 ft. from the foot of a cliff surmounted by an observa- \*~* T ion tower the angle of elevation of the top of the tower is 51° 25', and of 2-**}.^ the foot of the tower 31° 55'. Find the height of the cliff and of the tower. f*. How far from the foot of a flagpole 130 ft. high must an observer stand so that the angle of elevation of the top of the pole will be 25° ? 5. GA is a horizontal line, T is a point vertically above i; 5a point AG vertically below A. The angle BG A in minutes is Find Z BG T 4000 in degrees and minutes, given GA = 10,340 meters ; AT = 416.4 meters. 6. It is desired to find the height of a wireless tower situated on the top of a hill. The angle subtended by the tower at a point 250 ft. below" the base of the tower and at a distance measured horizontally of 2830 ft. from it is found to be 2° 42'. Find the height of the tower. 7. From a tower 428.3 ft. high the angles of depression of two objects tuated in the same horizontal line with the base of the tower and on the same side are 30° 22' and 47° 37'. Find the distance between them. 8. The summit of a mountain known to be 13,260 ft. high is seen at an angle of elevation of 27° 12' from a camp located at an altitude of 6359 ft. Compute the air-line distance from the camp to the summit of the mountain. 9. Two towns A and B, of which B is 25 miles northeast of A, are to be connected by a new road. 11 miles of the road is constructed from A in the direction N. 21° E. ; what must be length and direction of the remainder of the road, assuming it to be straight ? 22. Applications in Navigation. We shall confine ourselves to problems interne sailing; i.e. we shall assume that the dis- tances considered are sufficiently small so that the curvature of the earth may be neglected. II, § 22] THE RIGHT TRIANGLE 29 Definition. The course of a ship is the direction in which she is sailing. It is given either by the points of a mariner's compass (Fig. 21) as K E. by N. or in degrees and minutes ■ measured clockwise from the north. Observe that a " point " on a mariner's compass is 11° 15'. Hence for example, the course of a ship could be given either as N. E. by N. or as 33° 45 ; . A course S. E. by S. is the same as a course of 146° 15'. The distance a ship travels on a given course is always given Departure in nautical miles or knots. A knot is the length of a minute of arc on the earth's equator. (The earth's circumference is then 360 x 60 = 21,600 knots.) The horizontal component of the dis- tance is called the departure, the vertical com- ponent is called the difference in latitude. The departure is usually given in miles (knots), the difference in latitude in degrees and minutes. Fig. 22 Example. A ship starts from a position in 22° 12' N. lati- tude, and sails 321 knots on a course of 31° 15'. Find the difference in latitude, the departure, and the latitude of the new position of the ship, diff. in lat. = distance times cosine of course = 321 cos 31° 15' = 321 x 0.855 = 274' = 4° 34'. departure = distance times sine of course = 321 sin 31° 15' = 321 x 0.519 = 167 knots. Since the ship is sailing on a course which increases the lati- 30 PLANE TRIGONOMETRY [II, § 22 tude, the latitude of the new position is 22° 12' -f 4° 34' = 26° 46' N. Knowing the difference in latitude and the departure, we are able to calculate the new position of the ship, if the original position is known. In the preceding example, we found the latitude of the new position from the difference in latitude. To find the difference in longitude from the departure is not quite so simple. As the latitude increases, a given departure implies an increasing difference in longitude. Only on the equator is the departure of one nautical mile equivalent to a difference in longitude of one minute. The adjacent figure shows a departure AB in latitude <f>. The difference in longitude (in minutes) corresponding to AB is clearly the number of nautical miles in CD. Now arcs AB and CD are proportional to their radii PA and OC. Or, CD = ° C ~ . AB = A**-. (Why ?) PA cos<j> v J J In practice, it is customary to take for <f> in the determination of difference in longi- tude the so-called middle latitude, i.e. the latitude halfway between the original latitude and the final latitude. Thus in the preceding example, the original latitude was 22° 12' N, the final latitude was 26° 46' N. The middle lati- tude is therefore J (22° 12' + 26° 46') = 24° 29'. Hence ,-pp , , -, departure difference in longitude = . . , ,, — , — ■, — =r- cosme or middle latitude 167 167 = lg4 , m 30 4 , Fig. 23 cos 24° 29' 0.910 The determination of the position of a ship from its course and distance is known as dead reckoning. It is subject to con- siderable inaccuracy and must often in practice be checked by II, § 22] THE RIGHT TRIANGLE 31 direct determination of position by observations on the sun or stars. EXERCISES 1. A ship sails N. E. by E. at the rate of 12 knots per hour. Find the rate at which it is moving north. 2. A ship sails N. E. by N. a distance of 578 miles. Find its departure and difference in latitude. 3. A ship sails on a course of 73° until its departure is 315 miles. Find the actual distance sailed. Find also its difference in latitude. 4. A ship sails from latitude 47° \& N. 670 miles on a course N. W. by N. Find the latitude arrived at. 5. A ship sails from latitude 30° 24' N. and after 25 hours reaches lati- tude 35° 26' N. Its course was N. N. W. Find the average speed of the ship. 6. A vessel sails from lat. 24° 30' N., long. 30° 15 W., a distance of 692 miles on a course of 32° 20'. Find the latitude and longitude of its new position. 7. A vessel sails from lat. 10° 30' S., long. 167° 20' W., a distance of 692 miles on a course of 152° 30 f . Find the latitude and longitude of its new position. CHAPTEE III SIMPLE TRIGONOMETRIC RELATIONS /2Z. Other Trigonometric Functions. The reciprocals of ' the sine, the cosine, and the tangent of any angle are called, respectively, the cosecant, the secant, and the cotangent of that angle. Thus, cosecant = dlstance of P = - (provided y =#= 0). ordinate of P y , r. distance of P r , . , q , AN secant = — — : = - (provided x^=0). abscissa of P x f\ nsoissj-i Or r^ ^v cotangent = : — — — - (provided y ^= 0). ordinate of P p These functions are written esc 0, sec 0, ctn 0. From the definitions follow directly the relations esc 6= — , sec 8 = -, ctn 6 sin ' cos 9 ' tan 8 or esc • sin = 1, sec 6 • cos = 1, ctn • .tan = 1. To the above functions may be added versed sine (written versin), the co versed sine (written coversin), and the external secant (written exsec), which are defined by the equations versin = 1 — cos 0, coversin = 1 — sin 0, and exsec = sec — 1. Of importance in navigation and service- able in other applications (see § 88) is the haversine (written hav) which is defined to be equal to one half the versed sine ; i.e. have = |(1 — cos 0). r 24. The Representation of the Functions by Lines. Con- sider an angle in each quadrant and about the origin draw 32 Ill, § 24] SIMPLE TRIGONOMETRIC RELATIONS 33 Fig. 24 a circle of unit radius. Let P(x, y) be the point where the circle meets the terminal side of 6. Then sin 6 = ¥=zy, cos 6 = ^ = x, i.e. the sine is represented by the ordinate of P and the cosine by the abscissa. Hence the sine and cosine have respectively the same signs as the ordinate and abscissa of P. If we draw a tangent to the circle at the point A where the Fig. 25 circle meets the a^axis and let the terminal line of 9 meet this tangent in Q, we have tenO = ^Q = AQ, sec0 = -^2=OQ. Note that when 6 = 90°, 270°, and in general 90 + n . 360°, 270° + n • 360°, where n is any integer, there is no length AQ cut off on the tangent line and hence these angles have no tangents. If we draw a line tangent to the circle at the point B where D 34 PLANE TRIGONOMETRY [HI, § 24 the circle cuts the y-axis and let the terminal line of 6 cut this tangent in B, we have ctn0=z BK =B ^ and csc £ = OR = QR Fig. 26 EXERCISES 1. From Fig. 24 prove sin 2 + cos 2 = 1. 2. From Fig. 25 prove 1 + tan 2 = sec 2 0. I.' From Fig. 26 prove 1 + ctn 2 as csc 2 0. It Colon. A^ B*N /\\ / ^ 1 eK ^ \^ 25. Relations among the Trigonometric Functions. As one might imagine, the six trigonometric functions sine, cosine, tangent, cosecant, secant, cotangent are connected by certain relations. We shall now find some of these relations. From Fig. 9 (§ 10) it is seen that for all cases we have (1) x 2 + y 2 = r 2 . If we divide both sides of (1) by r 2 , we have + v- or sin 2 6 + cos 2 6 = 1. Dividing both sides of (1) by x 2 , we have 1 (by hypothesis r =£ 0) ; 1+ %=b ( if **°>- Therefore, 1 + tan 2 6 = sec 2 6. Similarly dividing both sides of (1) by y 2 gives or r ctn 2 6 + 1 + ! = -, (i*y*0); r csc 2 6. Ill, § 26] SIMPLE TRIGONOMETRIC RELATIONS 35 Moreover, we have y tane = ^ = :=^° x x cos 8 i r and, similarly, • cos 6 ctn6 = ^— S-. , sin 8 26. Identities. By means of the relations just proved any expression containing trigonometric functions may be put into a number of different forms. It is often of the greatest importance to notice that two expressions, although of a different form, are nevertheless identical in value. (How was an " identity" defined in algebra ?) The truth of an identity is usually established by reducing both sides, either to the same expression, or to two expres- sions which we know to be identical. The following examples will illustrate the methods used. Example 1. Prove the relation sec 2 + esc 2 = sec 2 esc 2 0. We may write the given equation in the form + -^— = sec 2 esc 2 0, cos 2 sin 2 sin 2 + cos 2 cos 2 sin 2 1 = sec 2 esc 2 0, = sec 2 esc 2 0, which reduces to cos 2 sin 2 sec 2 esc 2 = sec 2 esc 2 0. Since this is an identity, it follows, by retracing the steps, that the given equality is identically true. Both members of the given equality are undefined for the angles 0°, 90°, 180°, 270°, 360°, or any multiples of these angles. cos 2 Example 2. Prove the identity 1 4- sin — J 1 - sin Since cos 2 = 1 — sin 2 0, we may write the given equation in the form 1 + sin = 1 "" S1 " 2 9 or 1 + sin = 1 + sin 0. 1 - sin 36 PLANE TRIGONOMETRY [III, § 26 As in Example 1, this shows that the given equality is identically true. The right-hand member has no meaning when sin = 1 , while the left- hand member is defined for all angles. We have, therefore, proved that the two members are equal except for the angle 90° or (4 n-f 1)90°, where n is any integer. The formulas of § 25 may be used to solve examples of the type given in § 16. Example 3. Given that sin = ft and that tan is negative, find the values of the other trigonometric functions. Since sin 2 + cos 2 = 1, it follows that cos = ± Jf , but since tan is negative, lies in the second quadrant and cos0 must be — ||. More- over, the relation tan = sin 0/cos gives tan = — ft. The reciprocals of these functions give sec = — ||, esc = y, ctn — — *g. EXERCISES 1. Define secant of an angle ; cosecant ; cotangent. 2. Are there any angles for which the secant is undefined ? If so, what are the angles ? Answer the same question for cosecant and co- tangent. 3. Define versed sine ; co versed sine ; haversine. 4. Complete the following formulas : sin 2 6 + cos 2 = ? 1 + tan 2 = ? 1 + ctn 2 = ? tan = ? Do these formulas hold for all angles ? 5. In what quadrants is the secant positive ? negative ? the cosecant positive ? negative ? cotangent positive ? negative ? 6. Is there an angle whose tangent is positive and whose cotangent is negative ? 7. In what quadrant is an angle situated if we know that (a) its sine is positive and its cotangent is negative ? (b) its tangent is negative and its secant is positive ? (c) its cotangent is positive and its cosecant is negative ? — ' — *«••«£• Express sin 2 + cos so that it shall contain no trigonometric sA function except cos 0. 9. Transform (1 + ctn 2 0)csc so that it shall contain only sin 0. 10. Which of the trigonometric functions are never less than one in absolute value ? 11. For what angles is the following equation true : tan = ctn ? "^^wU- 12. How many degrees are there in when ctn = 1? ctn — — 1 ? sec = V2 ? esc = V£ ? ( H <^c3? Ill, § 27] SIMPLE TRIGONOMETRIC RELATIONS 37 13. Determine from a figure the values of the secant, cosecant, and cotangent of 30°, 150°, 210°, 330°. 14. Determine from a figure the values of the secant, cosecant, and cotangent of 45°, 135°, 225°, 315°. 15. Determine from a figure the values of the sine, cosine, tangent, secant, cosecant, and cotangent of 60°, 120°, 240°, 300°. 16. Find from the following equations. (a) sin0=£. (i) tan0= — 1. (6) sin = - \. (j) ctn = - 1. (c) cos = \. (k) tan = 1. (d) cos = - £. (I) ctn = 1. (e) sec = 2. (m) tan 2 = 3. (/) sec = - 2. (n) sin = 0. (gr) esc = 2. (b) cos = 0. (h) esc =•- 2. O) tan = 0. Prove the following identities and state for each the exceptional values of thjt variables, if any, for which one or both members are undefined : cos tan = sin0. sin ctn = COS0. 14 sin0 COS0 tor* ( H/»*— • C^r cos 1 — sin sin 2 — cos 2 = 2 sin 2 0-1. (1 — sin 2 0)csc 2 = ctn 2 0. tan + ctn = sec esc 0. [x sin + y cos 0] 2 4- [x cos — y sin 0] 2 = x 2 -f y 2 . 2^ =C os0. tan + ctn 1 — ctn4 = 2 esc 2 - esc 4 0. 26. tan 2 - sin 2 = tan 2 sin? 0. 27. 2(1 + sin 0) (1 + cos 0) = (1 + sin + cos 0) 2 28. sin 6 + cos« 0=1-3 sin 2 cos 2 esc esc 1- 27. The Trigonometric Functions of 90° -0. Figure 21 represents angles 6 antf 90° — 0, when is in each of the four X03 -(vr- ^A-v^ / (ooo -V y\>^~ uuSe*^ 38 PLANE TRIGONOMETRY [III, § 27 quadrants. Let OP be the terminal line of and OP' the terminal line of 90° - 0. Take OP' = OP and let (x, y) be Fig. 27 the coordinates of P and (x', y') the coordinates of P\ in all four figures we have x ' = y> y f = x > r' = r. Then Hence sin(9O°-0) = ^ = -: cos Also, cos (90° - 0) = - = 2 = sin 0, r r tan (90° -6) = ^ = -=ctn0. x' y esc (90° — 0)=sec0, sec (90° -0)= esc 0, ctn (90° -0)= tan 0. Definition. The sine and cosine, the tangent and cotangent, the secant and cosecant, are called co-functions of each other. The above results may be stated as follows : Any function of an angle is equal to the corresponding co-function of the com- plementary angle.* 28. The Trigonometric Functions of 180° — 6. By draw- ing figures as in § 27, the following relations may be proved : sin (180° - 6) = sin 0, esc (180° - 6) = esc 0, cos (180° - 0) = - cos 0, sec (180° - 0) = - sec 0, tan (180° - 0) = - tan 0, ctn (180° - 0) = - ctn 0. The proof is left as an exercise. * Two angles are said to be complementary if their sum is 90°, regardless of the size of the angles. Ill, § 29] SIMPLE TRIGONOMETRIC RELATIONS 39 29. The result of § 27 shows why it is possible to arrange the tables of the trigonometric functions with angles from 0° to 45° at the top of the pages and angles from 45° to 90° at the bottom of the pages. For example, since sin (90° — 0) = cos 0, the entry for cos will serve equally well for sin (90° — 6). As particular instances we may note sin 67° = cos 23°, tan 67° = ctn 23°, cos 67° = sin 23°. Verify these from the table. The result of § 28 enables us to find the values of the func- tions of an obtuse angle from tables that give the values only for acute angles. It will be noted that § 28 says that any function of an obtuse angle is in absolute value equal to the same function of its supplementary angle but may differ from it in sign. Thus to find tan 137° we know that it is in absolute value the same as tan (180° - 137°) = tan 43° = 0.9325. But tan 137° is negative. Hence tan 137° = - 0.9325. Similarly, sin 137°= 0.6820. cos 137° = - 0.7314. EXERCISES Find the values of the following : tan 146°, sin 136°, cos 173°, tan 100°, cos 96°, sin 138°, tan 98°, sin 145°, cos 168°, cos 138°, tan 173°, cos 157°. CHAPTER IV OBLIQUE TRIANGLES 30. Law of Sines. Consider any triangle ABC with the altitude CD drawn from the vertex C (Fig. 28). In all cases we have sin A Therefore, dividing, we obtain sin A a a = - , or sin B b sin A (i) (2) sin B If the perpendicular were dropped from B, the same argu- ment would give a/sin A = c/sin C. Hence, we have a b c sin A sin B sin C This law is known as the law of sines and may be stated as follows : Any two sides of a triangle are proportional to the sines of the angles opposite these sides. 31. Law of Cosines. Consider any triangle ABC with the altitude CD drawn from the vertex C (Fig. 29). In Fig. 29 a AD = b cos A ; CD = b sin A ; DB = c — b cos A. In Fig. 29 b AD = — b cos A ; CD = b sin A ; DB = c — b cos A. In both figures a2 = DB 2 -f CZ) 2 . 40 d IV, § 32] Therefore OBLIQUE TRIANGLES 41 a 1 = c 2 - 2 be cos A + b 2 cos 2 A + b 2 sin 2 A = c 2 — 2bc cos ^ + (cos 2 A + sin 2 ^1)6 2 , o c whence a 2 _ tf. + C 2 _ 2 be cos i4. The result holds also when A is a right angle. Why ? Similarly it may be shown that b 2 = c 2 + a 2 — 2 ca cos £, c 2 = a 2 + b 2 — 2 a6 cos C. Any one of these similar results is called the law of cosines. It may be stated as follows : Tlie square of any side of a triangle is equal to the sum of the squares of the other two sides diminished by twice the product of these two sides times the cosine of their included angle* 32. Solution of Triangles. To solve a triangle is to find the parts not given, when certain parts are given. From geometry we know that a triangle is in general determined when three parts of the triangle, one of which is a side, are given.f Eight triangles have already been solved (§ 15), and we shall now make use of the laws of sines and cosines to solve oblique triangles. The methods employed will be illustrated by some examples. It will be found advantageous to construct the triangle to scale, for by so doing one can often detect errors which may have been made. * Of what three theorems in elementary geometry is this the equivalent ? t When two sides and an angle opposite one of them are given, the triangle is not always determined. Why ? 42 PLANE TRIGONOMETRY [IV, § 33 33. Illustrative Examples. Example 1. Solve the triangle AB C, given = 276 A = 30° 20', B = 60° 45', a = 276. Solution : C = 180° - (A+ B) = 180° - 91° 5' = 88° 55'; : _ a sin B _ 270 sin 00° 45' = (270) (0*8725) = 476 9 . sin^l ' sin 30° 20' 0.5050 also c - ^_ sill _^ - 276 sin 88° 55' _ ( 276 ) (0.9998) __ 546 4 sin A sin 30° 20' 0.5050 "Check : It is left as an exercise to show that for these values we have c 2 = a 2 + b 2 — 2 ab cos C. Example 2. Solve the triangle ABC, given A = 30°, b = 10, a = 6. {? Constructing the triangle ABC, we see that two triangles AB X C and AB 2 C answer the descrip- * tion since b > a > altitude CD. Solution : Now ***! = *, or sin B, = ^^ =0.833, sin A a a whence B\ = 56°. 5. But B 2 = 180° - B x = 180° - 56°.5 = 123°.5, and Ci = 180° -{A + 50= 180° - 86°.5 = 93°. 5, C 2 = 180° - {A + ft) = 180° - 153°. 5 = 26 u .5. Now C2 _ sin C 2 or C2 _ a sin C 2 _ (6) (0.446) _ g 35 a sin ^1 ' sin -4 0.500 Also Ci = 8 inC 1 . or Ci= asinC 2 == (6)(0.998)_ 1198 a sin -4 ' sin J. 0.500 Check: Ci 2 = a 2 + & 2 — 2 ab cos 0i. 143.5 = 36 + 100 +(2) (6) (10) (0.061) = 143.3. C2 2 = a 2 + ^2 _ 2 a& cos C 2 . 28.62 = 36 + 100-(2)(6)(10)(0.895) = 28.60. IV, § 33] OBLIQUE TRIANGLES 43 Example 3. Solve the triangle ABC, given a = 10, 6 = 6, C = 40°. Solution : c 2 = a 2 + 6 2 — 2 ab cos (7 = 100 + 36 - (120) (0.766)= 44.08. Therefore c = 6.64. Now sin ^ = asinC = (10)(0.643) = c 6.64 ' i.e. A = 104°. 5. Likewise, sing = 6sinC = (6X0.643) = c 6.64 ' Check : A + B + C = 180°.0. Example 4. Solve the triangle ABC when C a = 7, 6 = 3, c = 5. From the law of cosines, &2 i C 2 _ a 2 i COSA= 26c =-, = -0.800, cos B = ?l±^l»! = 15 = 0.928, 2 ac 14 co S C = ?l+-^li? = 11 = 0.786. 2 06 14 i.e. B = 35° .5. s* = S JS* a = Fig. 7 33 Therefore .4 = 120°, Z* = 21°.8, C = 38°.2. Check : A + B + C = 180°.0 EXERCISES \£) Solve the triangle ABC, given /4 (a) ^1 = a0°, B = 70°, a = 100 ; (6) A = 40°, B = 70°, c = 110; xj\c) A = 45°.5, <7 = 68°.5, 6 = 40; \d) B=60°.5, C = 44°20', c = 20; e)Va = 30, & = 54, C = 50° ; ^ a = 10, 6 = 12, c = 14 ; 2Q 6 = 8, a = 10, C = 60° ; Jtf) a = 21, 6 = 24, c = 28. 2. Determine the number of solutions of the triangle ABC when (a) A = 30°, 6 = 100, a = 70 (6) 4 = 30°, 6 = 100, a = 100 (c) 21 = 30°, 6 = 100, a = 50 (d) 4 = 30°, 6 = 100, a = 40 (e) A = 30°, 6 = 100, a = 120 ; (/) J. = 106°, 6 = 120, a = 16 ; (gr) 4= 90°, 6= 15, a = 14. 44 PLANE TRIGONOMETRY [IV, 33 3. Solve the triangle ABC when (a) A = 37° 20', a = 20, 6 = 26 ; (c) 4 = 30°, a = 22, 6 = 34. (6; ^L = 37° 20', a = 40, 6 = 26; ( 4/^ h order to find the distance from a point A to a point B, a line -4C and the angles CAB and .A (72? were measured and found to be 300 yd., 60° 30', 5.6° 10' respectively. Find the distance AB. 5. In a parallelogram one side is 40 and one diagonal 90. The angle between the diagonals (opposite the side 40) is 25°. Find the length of the other diagonal and the other side. How many solutions ? 6. Two observers 4 miles apart, facing each other, find that the angles of elevation of a balloon in the same vertical plane with themselves are 60° and 40° respectively. Find the distance from the balloon to each observer and the height of the balloon. 7. Two stakes A and B are on opposite sides of a stream ; a third stake C is set 100 feet from A, and the angles A C B and CAB are observed to be 40° and 110°, respectively. How far is it from A to B ? 8. The angle between the directions of two forces is 60°. One force is 10 pounds and the resultant of the two forces is 15 pounds. Find the other force.* 9. Eesolve a force of 90 pounds into two equal components whose directions make an angle of 60° with each other. 10. An object B is wholly inaccessible and invisible from a certain point A. However, two points C and D on a line with A may be found such that from these points B is visible. If it is found that CD = 300 feet, AC = 120 feet, angle DCB = 70°, angle CDB - 50°, find the length AB. 11. Given a, 6, A, in the triangle ABC. Show that the number of possible solutions are as follows : A<90° f a < b sin A no solution, I b sin A < a < b two solutions, a>b one solution. | a = b sin A j ^^90° (a_6 no solution, a > b one solution. 12. The diagonals of a parallelogram are 14 and 16 and form an angle of 50°. Find the length of the sides. * It is shown in physics that if the line segments AB and AC represent in magnitude and direction two forces acting at a point A, then the diagonal AD of the parallelogram ABCD represents both in magnitude and direction the resultant of the two given forces. IV, § 34] OBLIQUE TRIANGLES 45 13. Resolve a force of magnitude 150 into two components of 100 and 80 and find the angle between these components. 14. It is sometimes desirable in surveying to extend a line such as AB in the adjoining figure. Show that this can be done by means of the broken line ABCDE. What measurements are necessary ? 15. Three circles of radii 2, 6, 5 are mutually tangent. Find the angles between their lines of centers. 16. In order to find the distance between two objects A and B on op- posite sides of a house, a station C was chosen, and the distances CA = 500 ft., CB = 200 ft., together with the angle ACB = 65° 30', were measured. Find the distance from A to B. 17. The sides of a field are 10, 8, and 12 rods respectively. Find the angle opposite the longer side. 18. From a tower 80 feet high, two objects, A and B, in the plane of the base are found to have angles of depression of 13° and 10° respec- tively ; thejiorizontal angle subtended by A and B at the foot C of the tower jedBP. Find the distance from A to B. Areas of Oblique Triangles. When tivo sides andjjie included angle are given. noting the area byQfjJire have from geometry 8 = i ch, but h = b sin A ; therefore (1) S = ±cbsmA. Likewise, S = i ab sin C and S = \ac sin B. Fig. &i 2. When a side and two adjacent angles are given. Suppose the side a and the adjacent angles B and C to be given. We have just seen that 8 = \ ac sin B. But from the law of sines we have a sin C sin A 46 PLANE TRIGONOMETRY [IV, § 34 Therefore Q_ a 2 ' sin B ♦ sin C 2 sin^t But sin A = sin [180° - (B + C)] = sin (5 + C). Therefore q _ a 2 sin jB sin (7 ^~ ~2 sin (B+C)' j 3. jWhen the three sides are given. ^*"W e have seen that S = \ be sin A. Squaring both sides of this formula and transforming, we have £2 = 2_1 sin 2 ^l = — (l-cos 2 ^l) 4 4 = 1(1 + 003.4). |(1- cos^); whence, by the law of cosines, 8*wmWl b * + c2 - a2 \ bcf 1 fr 2 + c 2 - a 2 ^ 2\ 26c y 2^ 26c J ^ 2&c + & 2 + c 2 -a 2 2 5c - b 2 - c 2 + a 2 4 ' 4 _6+_c_-j-a 5-f-c — a a—b + c ( a+J^c> ~ 2 * 2 ' 2 * " 2 which may be written in the form S 2 = s(s-a)(s-b)(s-c), where 2s = a + 6 + c. Therefore, (2) S = Vs(s -a)(s- b) (s-c). f 35^ The Radius of the Inscribed Circle. If r is the radius of-£ne inscribed circle, we have from elementary geometry, since s is half the perimeter of the triangle, S = rs ; equating this value of 8 to that found in equation (2) of the last article and then solving for r, we get, -v (s — a)(s — b)(s — c) s IV, § 36] OBLIQUE TRIANGLES 47 EXERCISES Find the area of the triangle ABC, given \> 1, a = 25, b = 31.4, C = 80° 25'. 4. a = 10, b = 7, C = 60°. 2- & = 24, c = 34.3, J. = 60° 25'. N» 5. a = 10, b = 12, C = 60°. 3. a = 37, 6 = 13, C = 40°. ^ 6. a = 10, 6 = 12, C = 8\ 7. Find the area of a parallelogram in terms of two adjacent sides and the included angle. 8. The base of an isosceles triangle is 20 ft. and the area is 100/V3 sq. ft. Find the angles of the triangle. Ans. 30°, 30°, 120°. \j 9. Find the radius of the inscribed circle of the triangle whose sides are 12, 10, 8. 10. How many acres are there in a triangular field having one of its sides 50 rods in length and the two adjacent angles, respectively, 70° and 60° ? and 60° \ 3,1 next The Law of Tangents. chapter the formulas in this and the next article will be needed. Let CD be the bisector of the angle G of the A ABC. Through A draw a line II DC, meeting BC produced in E. Then CE = b. Why ? From A draw a line q X DC meeting CB in F. At F draw a line r J_ AF meeting AB in G. AE=p. Now AACF is isosceles. Why? The angle ACE = ZA + /.B and the bisector of Z.ACE is _L CD. Hence Z CAF = Z CFA = ±Z(A + B). Moreover Z BAF= ZA-±Z(A + B) = ±Z(A-B). Let Now tan A + B and tan tan tan A + B A-B 48 PLANE TRIGONOMETRY [IV, § 36 But £ = !f = « + *. Why? tan Hence tan a, . Angles of a Triangle in Terms of the Sides. Con- f struct the inscribed circle of the triangle and. denote its radius by r.. If the perim- eter a + 6 + c = 2s, then (Fig. 36) AE = AF=s -a. BD = BF=s-b. CD=CE = s-c. . _ i ti r . ■ , >- r Then tan i ^4 = , tan \ B — , tan I C = s — a j s — b where, from § 35, rrUr^L /\ . = J (s-aXs-b)(s-c) _, A F tc £+£>/* VA A ->S-^i> + <WJ> M 38. Solution of Triangles by Means of the Haversine. The haversine may be used advantageously in the solution of triangles, (1) when two sides and the included angle are given ; (2) when the three sides are given. The law of cosines gives 2havJ. = 1 - cos^l = 1 - &2 + c2 - a2 2 6c _ q2_(fr _ c )2 2 be or 4 6chav A = a 2 — (b — c) 2 . 1. If 6, c and A are given we may find a from the formula (1) a 2 =(b-c) 2 + Ibch&vA. Similar formulas give b 2 or c 2 in terms of a, c, .B and a, 6, C respectively. 2. If a, 6, c are given, we may find A from the formula (2) hav A = *,-<»- «)' = .('-W-Q • w 4 be 6c Similar formulas will give B and 0, IV, § 38] OBLIQUE TRIANGLES 49 Example 1. Given A = 94° 23'.4, b = 55.12, c = 39.90. To find . By formula (1) above : 6 = 55.12 be = 2199 c = 39.90 hav 94° 23'.4 = 0.0446 (6-c) = 15.22 be hav A= 1184 (6-c)2 = 231.6 4 6c hav J. = 4736 4 be hav A = 4736 a* = 4968 » a = 70.49 Example 2. Given a = 4.51 , 6 = 6.13, c = 8.16. FindJL, B, C. a2 = 2034 hav ^ = 1«^ = 0.0811 A= 33*05 (6-c)2= 4.12 200.1 a 2_(6-c)2= 16.22 be = 50.02 4 6c = 200.1 62= 37.58 hav 5 = 2426 =0.1648 B = 47° 54 ( C _a)2= 13.32 147.21 62 _ ( C _ a )2 _ 24.26 ac= 36.80 4ac = 147.21 C 2= 66.59 hav C - 63,97 - 0.5785 C -- 99° 02' (6-a)2= 2.62 110.58 Check; 18QO , C 2«(6_ a )2 = 63.97 ab= 27.646 4 a& = 110.58 EXERCISES Solve the following triangles : ^ 1. a = 62.1, 6 = 32.7, c = 47.2. ^ 2. vl = 37°20', 6 = 2.4, c = 4.7. N 3. B = 121° 32', a = 27.9, c = 35.8. ^ 4. a = 3.2, 6 = 5.7, c = 6.5. 5. C = 72°21'.4, a = 314.1, 6 = 427.3. 6. a = 346.1, 6 = 425.8, c = 562.3. CHAPTER V 39. The Invention of Logarithms. The extensive numeri- cal computations required in business, in science, and in engi- neering were greatly simplified by the invention of logarithms by John Napier, Baron of Merchiston (1550-1617). By means of logarithms we are able to replace multiplication and division by addition and subtraction, processes which we all realize are more expeditious than the first two. If we consider the successive integral powers of 2 a) Exponent x 1 2 3 4 5 6 7 Result 2* . . 2 4 8 16 32 64 128 Exponent x 8 9 10 11 12 etc. A. P. Result 2* . . 256 512 1024 2048 4096 etc. G. P. we see that the results form a geometric progression (G. P.) and the exponents an arithmetic progression (A. P.). We know from elementary algebra that and x n x n Hence if we wish to multiply two numbers in our G. P. e.g. 4 x 8, we merely have to add the corresponding exponents 2. and 3 and under the sum 5 find the desired product 32. Sim- ilarly, if we wish to divide e.g. 4096 by 128, we merely have to subtract the exponent corresponding to 128, from that cor- 50 V, § 39] LOGARITHMS 51 responding to 4096 and under their difference 5 we find the desired quotient 32. To make the above plan at all useful it is evident that our table must be expanded so as to contain more numbers. First we can expand our table so that it will contain numbers less than 2, by subtracting 1 successively from the numbers in the A. P. and by dividing successively by 2 the numbers in the G.P. (2) In the second place we may find new numbers by inserting arithmetic means and geometric means. Thus, if we take the following portion of the preceding table -5 -4 -3 -2 -1 1 2 3 4 5 6 7 0.03125 0.0625 0.125 0.25 0.5 1 2 4 8 16 32 04 128 -2 - 1 1 2 3 4 * | 1 2 4 8 16 and insert between every two successive numbers of the upper line their arithmetic, and between every two successive num- bers of the lower line their geometric mean, we obtain the table (3) -2 -f -1 -i ± 1 1 2 5 3 i 4 i }V2 1 ^V2 1 V2 2 2V2 4 4V2 8 8V2 16 If the radicals are expressed approximately as decimals, this table takes the form -2.C -1. 5-l.< )-0.5 0.5 1.0 1.5 2 2.5 3 3.5 4 0.25 0.35 0.50 0.72 1.00 1.41 2.00 2.83 4.00 5.66 8.00 11.31 16 52 PLANE TRIGONOMETRY [V, § 39 By continuing this process we can make any number appear in the G. P. to as high a degree of approximation as we desire. To prepare an extensive table, which gives values at small inter- vals, is quite laborious. However, it has been done, and we have printed tables so complete that actual multiplication of any two numbers can" be replaced by addition of two other numbers. We shall soon learn how to use such tables. 40. Definition of the Logarithm. The logarithm of a number JV to a base b (b > 0, =£ 1) is the exponent x of the power to which the base b must be raised to produce the number JV. That is, if &*= N, then x ^lo&AT. These two equations are of the highest importance in all work concerning logarithms. One should keep in mind the fact that if either of them is given, the other may always be inferred. The numbers forming the A. P. in tables 1, 2, and 3 of § 39 are the logarithms of the corresponding numbers in the G. P., the base being 2. From table 3 we have 2* = 4 V2 which says log 2 4V2 = |. EXERCISES 1. When 3 is the base what are the logarithms of 9, 27, 3, 1, 81, |, 2. Why cannot 1 be used as the base of a system of logarithms ? 3. When 10 is the base what are the logarithms of 1, 10, 100, 1000 ? 4. Find the values of x which will satisfy each of the following equalities : (a) log 3 27 = x. (d) log a a = x. (g) log 2 x = 6. (6) \og x 3 = 1. (e) log a l=x. (h) log 32 z = |. (c) log, 5=|. (/) log, b \ = x. ('J) logo.001 x = 2. V, § 41] LOGARITHMS 53 5. Find the value of each of the following expressions : (a) log 2 16. (c) loge^ (e) log 25 125. (6) log 343 49. (d) log 2 Vl6. (/) log 2l fr. 41. The Three Fundamental Laws of Logarithms. From the laws of exponents we derive the following fundamental laws. I. TJie logarithm of a product equals the sum of the logarithms of its factors. Symbolically, log 6 MN = log 6 M + log 6 N. Proof. Let log 6 M = x, then b x = M. Let log 6 N= y, then 6 V = N. Hence we have MN = b x+y , or log 6 MN ax m + y, i.e. log 6 MN = log,, M + log 6 N. II. Tlie logarithm of a quotient equals the logarithm of the dividend minus the logarithm of the divisor. Symbolically, log 6 ^f= log 6 M - log & N. N Proof. Let log 6 M = x, then b* = M. Let log 6 N—y y then b' J m N. Hence we have M/N= b*'", or M M ^og b - = x-y, i.e. \og b ^. = \og b M - \og b N. III. The logarithm of the pth power of a number equals p times the logarithm of the number. Symbolically logfe M p = p log 6 M. Proof. Let log 6 M = x, then b x = M. Raising both sides to the pth power, we have b px = M v . Therefore log 6 M p =px=p log, M. Prom law III it follows that the logarithm of the real positive nth root of a number is one nth of the logarithm of the number. 54 PLANE TRIGONOMETRY [V, § 41 2 EXERCISES Given logi 2 = 0.3010, log 10 3 = 0.4771, logio 7 = 0.8451, find the of each of the f ollowingg|rxpressions : (a) log w 6. (/) logi 5. [Hint: logio 2x3= log 10 2 + logio 3.] [Hint: log 10 5 = log 10 y.J (6) logio 21.0. (?) logio m (c) logio 20.0. (h) logio Vl4. (d) logio 0.03! (i) logio 49^_ (e) logio |. (i) logio V24.7&. 2. Given the same three logarithms as in Ex. 1, find the value of each of the following expressions : r> * /„\ u„ 4 x 5 x 7 ,,x • ' 5 x 3 x 20 f * , 2058 (a) IogI °^2T^- (6) logl °-^T^- N(c) loSl0 ^i- ^(d) logio (2)*. (e) logic (3)8(5)«, (/) logio(2 3 )Q). <5> Logarithms to the Base 10. Logarithms to the base 10 are known as common or Briggian logarithms. Proceeding as in § 39 we can show that 10 - 3010 = 2, i.e. log 10 2 = 0.3010. Let ns multiply both members of the equation 10 03010 == 2 by 10, 10 2 , 10 3 , etc. and notice the effect on the logarithm. 10 o.3oio = 2 log 10 2 = 0.3010 10 3010 = 20 log 10 20 = 1.3010 10 2.3oio = 200 log L0 200 = 2.3010. It should be clear from this example that the decimal part of the logarithm (called the mantissa) of a number greater than 1 depends only on the succession of figures composing the num- ber and not on the position of the decimal point, ^vhile the in- tegral part (called the characteristic) depends simply on the position of the decimal point. Hence it is only necessary to tabulate the mantissas, for the characteristics can be found by inspection as the following considerations show. Since 10° = 1, lO^lO, 10 2 = 100, 10 3 = 1000, 10 4 = 10,000, etc. we have logj 1 = 0, log 10 10 = 1, log 10 100 = 2, log,o 1000 = 3, log M 10,000 = 4, etc. V, § 42] LOGARITHMS 55 It follows that a number with one digit (=f= 0) at the' left of the decimal point has for its logarithm a number equal to 4- a decimal ; a number with two digits at the left of its decimal point has for its logarithm a number equal to 1 -+- a decimal ; a number with three digits at the left of the decimal point has for its logarithm a number equal to 2 + a decimal, etc. We conclude, therefore, that the characteristic of the common loga- rithm of a number greater than 1 is one less than the number of digits at the left of the decimal point. Thus, logio 456.07 = 2.65903. The case of a logarithm of a number less than 1 requires special consideration. Taking the numerical example first con- sidered above, if log 10 2 =0.30103, we have log 10 0.2=0.30103-1. Why? This is a negative number, as it should be (since the logarithms of numbers less than 1 are all negative, if the base is greater than 1). But, if we were to carry out this subtraction and write log 10 0.2 = — 0.69897 (which would be correct), it would change the mantissa, which is inconvenient. Hence it is customary to write such a logarithm in the form 9.30103 - 10. If there are n ciphers immediately following the decimal point in a number less than 1, the characteristic is — n— 1. For convenience, ifn< 10, we write this as (9 — n) — 10. TJiis characteristic is written in two parts. The first part 9 — n is ivritten at the left of the ma?itissa and the — 10 at the right. In the sequel, unless the contrary is specifically stated, we shall assume that all logarithms are to the base 10. We may accordingly omit writing the base in the symbol log when there is no danger of confusion. Thus, the equation log 2 = 0.30103 means log 10 2 = 0.30103. To make practical use of logarithms in computation it is necessary to have a conveniently arranged table from which we can find (a) the logarithm of a given number and (b) the number corresponding to a given logarithm. The general ./ <* ) 56 PLANE TRIGONOMETRY [V, § 42 principles governing the use of tables will be explained by the following examples [Tables, pp. 110, 111]. Example 1. Find log 42.7. The characteristic is 1. In the column headed N (p. 110) we find 42 and if we follow this row across to the column headed 7, we read 6304, which is the desired mantissa. Hence log 42.7 = 1.6304. Example 2. Find log 0.03273. The characteristic is 8 — 10. The mantissa cannot be found in our table, but we can obtain it by a process called interpolation. We shall assume that to a small change in the number there corresponds a propor- tional change in the mantissa. Schematically we have u ' ^-' , Number Mantissa difference = 10 . T3270 -> 5145" L3273 -> ? 4 = difference 3280 — >» 5159 J Our desired mantissa is 5145 + ^-14 = 5149. Hence log 0.03273 = 8.5149 - 10. Example 3. Find x when log x — 0.8485. We cannot find this mantissa in our table, but we can find 8482 and 8488 which correspond to 7050 and 7060 respectively. Reversing the process of example 2, we have schematically Number Mantissa "7050 <- 84821 _" Difference = 10 ? <— 8485 J 6 = difference 7060 <- 8488 Hence the significant figures in our required number are 7050 -f- 1 • 10 = 7055. Since the characteristic is the required number is 7.055. EXERCISES /^) Find the logarithms of the following numbers from the table on ppYllO, 111 : 482, 26.4, 6.857, 9001, 0.5932, 0.08628, 0.00038. 2. Find the numbers corresponding to the following logarithms : 2.W35, 0.3502, 7.9599 - 10, 9.5300 - 10, 3.6598, 1.0958. 43. Use of Logarithms in Computation. The way in which logarithms may be used in computation will be suffi- ciently explained in the following examples. A few devices often necessary or at least desirable will be introduced. The V, § 43] LOGARITHMS 57 latter are usually self-explanatory. Reference is made to them here, in order that one may be sure to note them when they arise. The use of logarithms in computation depends, of course, on the fundamental properties derived in § 41. Example 1. Find the value of 73.26 x 8.914 x 0.9214. We find the logarithms of the factors, add them, and then find the number corresponding to this logarithm. The work may be arranged as follows : Numbers Logarithms 73.26 (-►) 1.8649 8.914 (-» 0.9501 0.9214 (-» 9.9645 - 10 12.7795 - 10 Product = 601.9 Arts. («-) 2.7795 Example 2. Find the value of 732.6 • 4- 89.14. Numbers Logarithms 732.6 (-*) 2.8649 89.14 (-*0 1.9501 Quotient = 8.219 Ans. («-) 0.9148 Example 3. Find the value of 89.14 -=- 732.6. Numbers Logarithms 89.14 c-*o 11.9501 - 10 732.6 (->) 2.8649 Quotient = 0.1217 Ans. «-) 9.0852 - 10 Example 4. Find the value of * x 21.63 Whenever an example involves several different operations on the logarithms as in this case, it is desirable to make out a blank form. When a blank form is used, all logarithms should be looked up first and entered in their proper places. After this has been done, the necessary opera- tions (addition, subtraction, etc.) are performed. Such a procedure saves time and minimizes the chance of error. Form Numbers Logarithms 763.3 (-►) 21.63 (-» ( + ) product 986.7 (-» (-)..... .... Ans. (<-) 58 PLANE TRIGONOMETRY [V, §43 Form Filled In Numbers Logarithms 763.2 (-» 2.8826 21.63 (-») 1.3351 product 4.2177 986.7 (-*) 2.9942 16.73 Ans. «-) 1.2235 Example 5. Find (1.357)5. Numbers Logarithms 1.357 (-» 0.1326 (1.357)5 = 4.602 Ans. (-*-) 0.6630 Example 6. Find the cube root of 30.11. Numbers Logarithms 30.11 (-» 1.4787 #30.11 =3.111 Ans. («-) 0.4929 Example 7. Find the cube root of 0.08244. Numbers Logarithms 0.08244 (->) 28.9161 - 30 #0.08244 = 0.4352 Ans. («<-) 9.6387-10 6 D EXERCISES Compute the value of each of the following expressions using the table on pp. 110, 111. 1. 34.96 x 4.65. 2. 518.7 x 9.02 x .0472. 3 0.5683 _ 0.3216* 4 5.007 x 2.483 6.524 x 1.110* 5. (34.16 x .238)2. 6. 8.572 x 1.973 x (.8723)2. K.# 8076 x 3.184 (2.012)5 O 10. a/ 2941 >< 17 - 11. '2173 x 18.75 #0.00732 #735 ^12. (20.027)* d/lS. 2 1( ». ■' i e 648.8 '(21.4)1 / 1379 >2791 " / v 14. Vio^.ioo 2 . 15. (0.02735)*. die. f A #3275 (2.01) y i V, § 44] LOGARITHMS 59 44. Cologarithms. Since — and M • — - are equivalent, we may in a logarithmic computation, add the logarithm of — instead of subtracting log N. The logarithm of -— is called the cologarithm of N. Therefore colog N = log 1/N = log 1 — log N = — log N, since log 1 is zero. We write cologarithms, like logarithms, with positive man- tissas. Therefore the cologarithm is most easily found by sub- tracting the logarithm from zero, written in the form 10.0000 -10. Example. Find the colog 27.3. 10.0000 - 10 i log 27. 3= 1.4362 colog 27.3= 8.5638-10 The cologarithm can be written down immediately by subtracting the last significant figure of the logarithm from 10 and each of the others from 9. If the logarithm is positive the cologarithm is negative and hence — 10 is affixed. There is no gain in using cologarithms when we have a quotient of two numbers. There is an advantage when either the numerator or denomi- nator contains two or more factors, for we can save an operation of addi tion or subtraction. Let us solve Ex. 4, § 43, using cologarithms. Example. Find the value of 763 ' 2 x 2L63 ■ 986.7 Numbers Log 763.2 .->■ 2.8826 21.63 -> 1.3351 986.7 -> (colog) 7.0058 - 10 16.73 <- 1.2235 EXERCISES Compute the value of each of the following expressions, using cologa- rithms. /?\ J 2.80 x 37.6 /"T\ J 97.63 x 876.5 2876 x3.4 x 2.987 60 PLANE TRIGONOMETRY [V, §44 3 5 5 V3275 , ' 7 x 8 x 9 x 27.6 ^J (2.01)*(1.76)» 4. 312 • 6 1293 x 12 7 x 5 610,27 N ^(l + 2 V3)(760 + 8)' MISCELLANEOUS EXERCISES 1. What objections are there to the use of a negative number as the base of a system of logarithms ? 2. Show that a l °s a x = x. 3 . Write each of the following expressions as a single term : 'a) log x + log y — log z. QpS^P log x — 2 log y + 3 log z. XcpS log a — log (x + y) - \ log (ex + tf)'+ log Vw + x. 4} Solve for x the following equations : §2 log 2 £ + log 2 4 = 1. (c) 2 logio x - 3 log 10 2 = 4. log 3 x - 3 log 3 2 = 4. (d) 3 log 2 x + 2 log 2 3 = 1, - /5. How many digits are there in 2 35 ? 3 142 ? 3 12 x 2» ? ^g| y6. Which is the greater, (f£) 100 or 100 ? / 7> Find the value of each of the following expressions : (a% log 6 35. ((py log 3 34. (g) log 7 245. (d) log 13 26. - 8. Prove that log b a • log a 6 = 1. 9. Prove that log„ a; + V x2 ~ - = 2 lo go [x + Vx 2 - 1]. « — Vx 2 — 1 10. The velocity v in feet per second of a body that has fallen s feet is given by the formula v = V64.3s. What is the velocity acquired by the body if it falls 45 ft. 7 in. ? / 11. Solve for x and ?/ the equations ; 2 X = 16v, x + 4 ?/ = 4. ► m CHAPTER VI LOGARITHMIC COMPUTATION 46. Logarithmic Computation. In the last chapter a few examples of the use of logarithms in computation were given in connection with a four-place table. Such a table suffices for data and results accurate to four significant figures. When greater accuracy is desired we use a five-,' six-, or seven-place table. No subject is better adapted to illustrate the use of logarith- mic computation than the solution of triangles, which we shall consider in some detail. Five-place tables and logarithmic solutions ordinarily are used at the same time, since both tend toward greater speed and accuracy. 46. Five-place Tables of Logarithms and Trigonometric Functions. The use of a five-place table of logarithms differs from that of a four-place table in the general use of so-called " interpolation tables " or " tables of proportional parts," to facilitate interpolation. Since the use of such tables of pro- portional parts is fully explained in every good set of tables, it is unnecessary to give such an explanation here. It will be assumed that the student has made himself familiar with their use.* In" the logarithmic solution of a triangle we nearly always need to find the logarithms of certain trigonometric functions. For example, if the angles A and B and the side a are given, we find the side b from the law of sines given in § 30, , _ a sin B ♦^* sin A * For this chapter, such a five-place tahle should be purchased. See, for example, The Macmillan Tables, which contain all the tables mentioned here with an explanation of their use. 61 62 PLANE TRIGONOMETRY [VI, § 46 To use logarithms we should then have to find log a, log (sin B) and log (sin A). With only a table of natural functions and a table of logarithms at our disposal, we should have to find first sin A, and then log sin A. For example, if A = 36° 20', we would find sin 36° 20' = 0.59248, and from this would find log sin 36° 20' = log 0.59248 m 9.77268 - 10. This double use of tables has been made unnecessary by the direct tabulation of the logarithms of the trigonometric functions in terms of the angles. Such tables are called tables of logarithmic sines, logarithmic cosines, etc. Their use is explained in any good set of tables. The following exercises are for the purpose of familiarizing the student with the use of such tables. J EXERCISES V. Find the following logarithms : * (a) log cos 27° 40'.5. (d) log ctn 86° 53'. 6. (6) log tan 85° 20'.2. (e) log cos 87° 6'.2. \}c) log sin 45° 40'. 7. (/) log cos 36° 53'. 3. "■■k. Find A, when (a) log sin A = 9.81632 - 10. (d) log sin A = 9.78332 - 10. • (6) log cos A = 9.97970 - 10. (e) log ctn } A = 0.70352. (c) log tan A = 0.45704. •(/) log tan \A = 9.94365 - 10. VL Find Mf tan fl = 476 - 32 x 89 - 710 . \ 87325 ^ 4. Given a triangle ABC, in which ZA = 32°, Z B = 27°, a = 5.2, find b by use of logarithms. 47. The Logarithmic Solution of Triangles. The effective use of logarithms in numerical computation depends largely on a proper arrangement of the work. In order to secure this, the arrangement should be carefully planned beforehand by constructing a blank form, which is afterwards filled in. More- over, a practical computation is not complete until its accuracy has been checked. The blank form should provide also for a good check. Most computers find it advantageous to arrange * Five-place logarithms are properly used when angles are measured to the nearest tenth of a minute. For accuracy to the nearest second, six places should be used. VI, § 48] LOGARITHMIC COMPUTATION 63 the work in two columns, the one at the left containing the given numbers and the computed results, the one on the right containing the logarithms of the numbers each in the same horizontal line with its number. The work should be so arranged that every number or logarithm that appears is properly labeled ; for it often happens that the same number or logarithm is used several times in the same computation and it should be possible to locate it at a glance when it is wanted. The solution of triangles may be conveniently classified under four cases : Case I. Given two angles and one side. Case II. Given two sides and the angle opposite one of the sides. Case III. Given two sides and the included angle. Case IV. Given the three sides. In each case it is desirable (1) to draw a figure representing the triangle to be solved with sufficient accuracy to serve as a rough check on the results ; (2) to write out all the formulas needed for the solution and the check ; (3) to prepare a blank form for the logarithmic solution on the basis of these formulas ; (4) to fill in the blank form and thus to complete the solution. We give a sample of a blank form under Case I ; the student should prepare his own forms for the other cases. 48. Case I. Given Two Angles and One Side. Example. Given: a=430.17, ^1=47° 13'.2, B=52° 29'.5. (Fig. 37.) To find: C, 6, c. Formulas : C = 180°-(A + B), b=—2-sinB, sin A sin C. sin A Check (§ 36): ^=± = tan $(<?-*) . ^ J c + b tanJ(C+B) * Fig. 37 64 PLANE TRIGONOMETRY [VI, § 48 The following is a convenient blank form for the logarithmic solu- tion. The sign (+) indicates that the numbers should be added ; the sign (— ) indicates that the number should be subtracted from the one just above it. A = ( + )* = A+ B = C = a = sin A = Numbers 179° 60'.0 Logarithms sin a/sin A sin B = sin b = . . a/sin A sin C c c-b = c+ b = C-B=. . C+ B= . . tan | ( C — B) = tan tan \{C + B)= tan (-) -») ( + ) -H (+) Check ■» (-) •) (-) (1) (Logs (1) and (2) . should be equal . for check.) "(2) Filling in this blank form, we obtain the solution as follows. Numbers A= 47°13'.2 B= 52°29'.6 A+ B= 99°42'.8 179° 60'.0 Logarithms 0= 80°17'.2 a^= 430.17 sin A =sin47°13'.2 a/sin A sin B = sin 52° 29'. 6 b = 464.94 Ans. 2.63364 (-) 9.86567 - 10 2.76797 ( + ) 9.89943 - 10 2.66740 Check* VI, § 49] LOGARITHMIC COMPUTATION 65 a/sin A 2.76797 sin C = sin 80° 17'. 2 (->) ( + ) 9.99373 - 10 c = 577.70 Ans. (<-) 2.76170 Check c-b = 112.76 (->) 2.05215 c + b = 1042.64 (->) (-) 3.01813 9.03402 - 10 C-B = 27°47'.6 C+£ = 132°46'.8 tan|(C- i*)=tanl3°53'.8 (->) 9.39342-10 tan£(C + 5)= tan 66° 23'. 4 (->-) (-) 0.35942 9.03400 - 10 EXERCISES Solve *ud ulUWft the following triangles ABC : . V. a = 372.5, ^4 = 25° 30', 5 = 47° 50'. >* X c = 327.85, A = 110° 52'.9, 5 = 40° 31'.7. Ans. C = 28° 35'.4, a = 640.11, 6 = 445.20. 3. a = 53.276, A = 108° 50'.0, C = 57° 13'. 2. ^ V b = 22.766, B = 141° 59M, C = 25° 12'.4. 5. b = 1000.0, B = 30° 30'.5, C = 50° 50'.8. X, «' a = 257.7, J. = 47° 25', B = 32° 26'. 49. Case n. Given Two Sides and an Angle Opposite One of Them. If A, a, b are given, B may be determined from the relation (1) AnB = bsmA - a If log sin B = 0, the triangle is a right triangle. Why ? If log sin B > 0, the triangle is impossible. Why ? If log sin B < 0, there are two possible values, B u B 2 of 5, which are supplementary. Hence there may be two solutions of the triangle. (See Example.) No confusion need arise from the various possibilities if the corresponding figure is constructed and kept in mind. It is desirable to go through the computation for log sin B * A small discrepancy in the last figure need not cause concern. Why ? 66 PLANE TRIGONOMETRY [VI, § 49 before making out the rest of the blank form, unless the data obviously show what the conditions of the problem actually Example L Given : A = 46° 22'.2, a = 1.4063, b = 2.1048. (Fig. 38.) To find: B, C, c. Formula : sin B = bsinA . Fig. 38 Numbers Logarithms 6 = 2.1048 (->) 0.32321 sin A = sin 46° 22' .2 (->) ( + ) 9.85962 -10 bsinA 0.18283 a =1.4063 (->) (-) 0.14808 sin B (-<-) 0.03475 Hence the triangle is impossible. Why ? Example 2. Given : a = 73.221, b = 101.53, A = 40° 22'.3. (Fig. 39.) To find : B, C, c. Formula: sin£= &sin ^ . Numbers Logarithms b = 101.53 (->*) 2.00660 sin ^L= sin 40° 22'. 3 (->) ( + ) 9.81140 - 10 6sin^i a = 73.221 sin i? 11.81800-10 (->*) (-) 1.86464 9.95336 - 10 The triangle is therefore possible and has two solutions (as the figure shows) . We then proceed with the solution as follows : We find one value 2?i of B from the value of log sin B. The other value B 2 of B is then given by B 2 = 180° - B x . VI, § 49] LOGARITHMIC COMPUTATION 67 Other formulas : C= 180° -(A + B). a sin C sin A Check: ^^ c + b _tanKC-B) < tan£(C + B) Numbers Logarithms sin B 9.95336 - 10 i*i= 63° 55'. 2 179° 60\0 B 2 = 116° 4' .8 A + B x = 104° 17'.5 179° 60'.0 d= 75°42'.5 a (->) 1.86464 sin .4 (-►) (-) 9.81140-10 a/ahiA 2.05324 sin d = sin 75° 42'. 5 (-►) ( + ) 9.98634 - 10 d = 109.54 O-) 2.03958 d-b= 8.01 (->) 0.90363 ci + 6 = 211.07 (->) (-) 2.32443 8.57920 - 10 C l -B l = 11°47'.3 Ci + Bi = 139° 37 '.7 tan 4(Ci— JBi)= tan 5° 53'. 6 (-►) 9.01377 - 10 tan K Ci + -Bi) = tan 69° 48' . 8 (-*►) 0.43455 8.57922 - 10 \ Check. One solution of the triangle gives, therefore, B = 63° 55'. 2, C = 75° 42'. 5, c = 109.54. To obtain the second solution, we begin with B 2 = 116° 4'. 8. We find C 2 from C 2 = 180° - (A + B 2 ); i.e. C 2 = 23° 32'. 9. The rest of the com- putation is similar to that above and is left as an exercise. EXERCISES 1. Show that, given J., a, 6, if A is obtuse, or if J. is acute and a > 6, there cannot be more than one solution. Solve the following triangles and check the solutions : J 2. a = 32.479, 6 = 40.176, A = 37° 25M. 68 PLANE TRIGONOMETRY [VI, § 49 /: 3. 6 = 4168.2, 4. a = 2.4621, 5. a = 421.6, 6. a = 461.5, 3179.8, 4.1347, 532.7, c = 121.2, B = 51°21'A. B = 101° 37'.3. A = 49° 21 '.8. C=22°31'.6. 7. Find the areas of the triangles in Exs. 2-5. 50. Case III. Given Two Sides and the Included Angle. Example. Given: a=214.17, 6=356.21, B C = 62° 21 '.4. (Fig. 40.) / N. To find: A, B, c. V ^v Formulas : V \ tan| (B-A)= l L=Jtt^l(B + A); p> \ B + A = 180° - O = 117° 38'.6 a sin C C b = 356.Sl J. Fig. 40 sin J. Numbers Logarithms 6 - a = 142.04 c-*o 2.15241 6 + a = 570.38 (-» (-) 2.75616 (6 - a)/(b + a) 9.39625 - 10 tan |(1? + A) = tan 58° 49'.3 (-» ( + ) 0.21817 tan^(£- A)= tan 22° 22'. 2 («-) 9.61442 - 10 .-. J.= 36°27'.l .Ans. 2*= 81° 11'.5 J.W8. a = 214.17 (— ►) 2.33076 sin^L = sin36°27'.l (-» (-) 9.77389- 10 a/sin .4 2.55687 sin C = sin 62° 21'. 4 (-» ( + ) 9.94736-10 c = 319.32 .4ns. («-) 2.50423 Check by finding log (6/sin B). I SXERCI SES f Solve and check each of the following triangles : 1. a = 74.801, 6 = 37.502, C = 63°35'.5. ^ 2. a = 423.84, 6 = 350.11, G = 43° 14'.7. -s 3. 6 = 275, c = 315, A = 30° 30/. 4. a = 150.17, c = 251.09, B = 40°40'.2 ; > 6. a = 0.25089, 6 = 0.30007, C = 42° 30' 20". 6. Find the areas of the triangles in Exs. 1-5. VI, § 51] LOGARITHMIC COMPUTATION 69 51. Case IV. Given the Sides. Example. Given: a = 261.62, 6 = 322.42, c = 291.48. To find: A, B, C. Formulas : s = K« + b + c). r _J(« — a) — 6) (8 — c) r-yj 8 tan i A = r -, tani£ = -^-, s- a « — 6 Check : A + B + C = 180°. Numbers a = 261.62 6 = 322.42 c = 291.48 < 28 = 875.52 8 = 437.76 8 — a = 176.14 8 - b = 115.34 8- - c = 146.28 tan \C- s — c Logarithms (-►) s = 437.76 (Check). *(- 2.24586 2.06198 ) ( + ) 2.16518 6.47302 ) (-) 2.64124 3.83178 r s 1 — a tan | A - tan 25° 4'. 1 r s-b tan£ Bz= tan35°32'.4 r = s — c = («-) «-) («-) 1.91589 2.24586 9.67003 - 10 1.91589 2.06198 9.85391 - 10 191589 2.16518 9.75071 - 10 A= 50° 8'.2 Ans. B= 71° 4'.8 ^?is. C = 58° 46'.9 ^Ins. 179° 59'.9 (Check.) "7Vys> *By adding 8— a, 8 — 6, s (A-*)* r^-t (ft- e}« t^. ^ fc A (§37) 70 PLANE TRIGONOMETRY [VI, § 51 EXERCISES Solve and check each of the following triangles : VI. a as 2.4169, b = 3.2417, c = 4.6293. *%!.<*= 21.637, & = 10.429, c = 14.221. 5. a as 528.62, . 6 = 499.82, c = 321.77. 4. a = 2179.1, 6 = 3467.0, c = 5061.8. V« a = 0.1214, & = 0.0961, c = 0.1573. 6. Find the areas of the triangles in Exs. 1-5. 7. Find the areas of the inscribed circles of the triangles in Exs. 1-5. OTHER LOGARITHMIC COMPUTATIONS 52. Interest and Annuities. Simple Interest. Let the principal be represented by P the interest on $ 1 for one year by r the number of years by n the amount of P for n years by A n Then the simple interest on P for a year is Pr the amount of P for a year is P + Pr =P (1-4- r), the simple interest on P for n years is Pnr the amount of P for n years is A n =P(1 + nr). Example. How long will it take $210, at 4% simple interest, to amount to $ 298.20 ? A n = P(l + nr) i.e. n = An ~ P . Pr Number Logarithm A n - P = 88.20 ->- 1.9455 Pr= 8.40 -^ 0.9243 n = 10.5 -«— 1.0212 10 yr. 6 mo. ^Ins. Compound Interest. Let the original principal be P and the rate of interest r Then the amount A] at the end of the first year is A x = P-hPr=:P(l-\-r), VI, § 52] LOGARITHMIC COMPUTATION 71/ the amount A 2 at the end of the second year is A 2 = A 1 (l + r) = P(l + ry, the amount at the end of n years is 4,«J»(l+r)". If the interest is compounded semiannually, A n -— pf 1 + M , 1+-) , if q times a year^l n =P( 1 + - j ■ Since P in n years will amount to A H , it is evident that P at the present time may be considered as equivalent in value to A due at the end of n years. Hence P is called the present worth of a given future sum A. Since A n = P(l + r)% P= A n (1 + r)"\ Example. In how many years will one dollar double itself at 4 % in- terest compounded annually ? A n = P(\ + r) - or log ^ = nlog(l + r). . n = logA-log-P log (1 + r) Hence n = log2 - log 1 = 0,3010 = 17 . 7 . log (1.04) 0.0170 17 yr. 9 mo. Ans. Annuities. An annuity is a fixed sum of money payable at equal intervals of time. To find the present worth of an annuity of A dollars pay- able annually for n years, beginning one year hence, the rate of interest being r and the number of years n. Since the present worth of the first payment is A (1 + r) _1 , of the second A(l -f- r) -2 , etc., the present worth of the whole is P=^[(l + r)-i+(l-f r)-*+ .- +(l + r)-*]. The quantity in the brackets is a G. P. whose ratio is (1 + r)~K Summing, we have l-(l + r)-i r\_ {1 + ryj 72 PLANE TRIGONOMETRY [VI, § 52 If the annuity is perpetual, i.e. n is infinite, the formula for A present worth becomes P -— — • Example. What should be paid for an annuity of $ 100 payable an- nually for 20 years, money being worth 4 % per annum ? p=Mh LLl. 0.04 L (1.04) 20 J 20 = 2.188. Therefore P= — fl L-1 =2500 f U^§1 =$1358, approximately. 0.04 L 2.188 J L2.188J ' FF J (1.04) By logarithms (1 .04) 20 - 2. 188. 53. Projectiles. Logarithms are used extensively in ballis- tic computations. [Ballistics is the science of the motion of a projectile.] The following is a very simple example of the type of problem considered. The time of flight of a projectile (in vacuum) is given by the formula T=\- * where X is the horizontal range * 9 in feet, <f> is the angle of departure, and g is the acceleration due to gravity in feet per second per second \_g — 32.2]. If it is known that the range is 3000 yd. and that the angle of de- parture is 30° 20', find the time of flight. T /2Xtan<£ " X 9 Numbers Logarithms 21= 18000 ~* 4.2553 tan 30° 20' -* 9.7673 - 10 4.0226 32.2 "* 1.5079. 2)2.5147 18.09 <— 1.2574 T = 18.09 seconds. Ans. EXERCISES 1. Find the amount of $ 500 in 10 years at 4 per cent compound inter- est, compounded semiannually. 2. In how many years will a sum of money double itself at 5 per cent interest compounded annually ? semiannually ? VI, § 54] LOGARITHMIC COMPUTATION 73 3. A thermometer bulb at a temperature of 20° C. is exposed to the air for 15 seconds, in which time the temperature drops 4 degrees. If the law of cooling is given by the formula = doe- 61 , where 6 is the final tem- perature, #o the initial temperature, e the natural base of logarithms, and t the time in seconds, find the value of b. 4. The stretch s of a brass wire when a weight m is hung at its free end is given by the formula j 8 = — — , where m is the weight applied in grams, g = 980, I is the length of the wire in centimeters, r is the radius of the wire in centimeters, and fc is a constant. If m = 844.9 grams, I = 200.9 centimeters, r = 0.30 centi- meter when s = 0.056, find k. 5. The crushing weight P in pounds of a wrought-iron column is given by the formula ,73.55 P= 299,600^—, p where d is the diameter in inches and I is the length in feet. What weight will crush a wrought-iron column 10 feet long and 2.7 inches in diameter? 6. The number n of vibrations per second made by a stretched string is given by the relation 2 rz-r- n = 2TV^r' where I is the length of the string in centimeters, M is the weight in grams that stretches the string, m the weight in grams of one centimeter of the string, and g = 980. Find n when M = 5467.9 grams, I = 78.5 centimeters, m = 0.0065 gram. 7. The time t of oscillation of a pendulum of length I centimeters is given by the formula ,— — >(980 Find the time of oscillation of a pendulum 73.27 centimeters in length. 8. The weight w in grams of a cubic meter of aqueous vapor saturated at 17° C. is given by the formula = 1293 x 12.7 x 5 (1 + ^X760x8)* Compute w. 54. The Logarithmic Scale. An arithmetic scale in which the segments from the origin are proportional to the logarithms of 1, 2, 3, etc., is called a logarithmic scale. Such a scale is given in Fig. 42. i I JIIIJ1 Fig. 42 74 PLANE TRIGONOMETRY [VI, § 55 55. The Slide Rule. The slide rule consists of a rule along the center of which a slip of the same material slides in a groove. Along the Fig. 43 upper edge of the groove are engraved two logarithmic scales, A and B, that are identical. Along the lower edge are also two identical logarithmic scales, and D, in which the unit is twice that in scales A and B. Since the segments represent the logarithms of the numbers found in the scale, the operation of adding the segments is equivalent to multiplying the f 1 2 £ * 1 4 I ) 6 7 i 5 9 1 2 A 1 ! 1 1 Ml ii i I I I ! M l 1 1 mil ilmlilililililililili iiilii 1,1,1 1 J EH III IIJJ 11,1,1 ,1, jTI'II WMV-, \ r oL 1 II IK II II III But 1 JIIIIJII \ B r \ !' ! i \ D l 2 3^ 4 5 6 7 8 9 ) rl ■ ■ 2 3 / I C- 1 1 ll Hill llllll II M'lll Ii IjlJ ll tiin TtT&U n, INI INI IIIIIHIIHI HI nil I If MM 1 J.-l 1.1 . 1J ftf4 .njijl 1 2 3 4 Fig. 44 corresponding numbers. Thus in Fig. 44 the point marked 1 on scale B is set opposite the point marked 2.5 on scale A. The point marked 4 on scale B will be opposite the point marked 10 on scale A, i.e. 2.5 x 4 = 10. Similarly we read 2.5 x 3.2 = 8, 2.5 x 2.5 = 6.25. Other multiplications can be performed in an analogous manner. Division'can be performed by reversing the operation. Thus in Fig. 44 every number of scale B is the result of dividing the number above it by 2.5. Thus we read 7.2 -~ 2.5 = 2.9 approximately. Since scales G and D are twice as large as scales A and B, it follows that the numbers in these scales are the square roots of the numbers opposite to them in scales A and B. Conversely the numbers on scales A and B are the squares of the numbers opposite them on scales C and D. Moreover the scales C and D can be used for multiplying and divid- ing, but the range of numbers is not so large. For a more complete discussion of the use of a slide rule consult the book of instructions published by any of the manufacturers of slide rules, where also exercises will be found for practice. CHAPTEK VII TRIGONOMETRIC RELATIONS 56. Radian Measure. In certain kinds of work it is more convenient in measuring angles to use, instead of the degree, a unit called the radian. A radian is defined as the angle at the center of a circle whose subtended arc is equal in length to the radius of the circle (Fig. 45). Therefore, if an angle $ at the center of a circle of radius r units subtends an arc of s units, the measure of 6 in radians is r Since the length of the whole circle is 2 -n-r, it follows that — = 2tt radians = 360°, r or (2) it radians = 180°. Therefore, 180° TT 1 radian = = 57° 17' 45" (approximately). FlG 45 It is important to note that the radian * as defined is a con- stant angle, i.e. it is the same for all circles, and can therefore be used as a unit of measure. From relation (2) it follows that to convert radians into degrees it is only necessary to multiply the number of radians by 180/7T, wliile to convert degrees into radians we multiply the number of degrees by tt/180. Thus 45° is tt/4 radians ; 7r/2 radians is 90°. * The symbol r is often used to denote radians. Thus 2 r stands for 2 radians, ir r for tt radians, etc. When the angle is expressed in terms of it (the radian being the unit), it is customary to omit r . Thus, when we refer to an angle it, we mean an angle of it radians. When the word radian is omitted, it should be mentally supplied in order to avoid the error of supposing ir means 180. Here, as in geometry, t = 3.14159. . . . 75 76 PLANE TRIGONOMETRY [VII, § 57 57. The Length of Arc of a Circle. From relation (1), § 56, it follows that s = r8. That is (Fig. 46), if a central angle is measured in radians, and if its intercepted arc and the radius of the circle are measured in terms of the same unit, then length of arc = radius x central angle in radians. r~ EXERCISES 1. Express the following angles in radians : 25°, 145°, 225°, 300°, 270°, 450°, 1150°. -* 2. Express in degrees the following angles : ■K 7 IT blT 5-TT — , — , , u 7T, . 4' 6 6 '4 * 3. A circle has a radius of 20 inches. How many radians are there in an angle at the center subtended by an arc of 25 inches ? How many degrees are there in this same angle ? Ans. | r ; 71° 37' approx. — i 4. Find the radius of a circle in which an arc 12 inches long subtends an angle of 35°. """ 5. The minute hand of a clock is 4 feet long. How far does its ex- tremity move in 22 minutes ? 6. In how many hours is a point on the equator carried by the rotation of the earth on its axis through a distance equal to the diameter of the earth? 7. A train is traveling at the rate of 10 miles per hour on a curve of half a mile radius. Through what angle has it turned in one minute ? 8. A wheel 10 inches in diameter is belted to a wheel 3 inches in diameter. If the first wheel rotates at the rate of 5 revolutions per \\g minute, at what rate is the second rotating? How fast must the former rotate in order to produce 6000 revolutions per minute in the latter ? 58. Angular Measurement in Artillery Service. The divided circles by means of which the guns of the United States Field Artillery are aimed are graduated neither in degrees nor in radians, but in units called mils. The mil is defined as an angle subtended by an arc of ^^-q of the circumference, and is therefore equal to 2tt 3.1416 6400 3200 0.00098175 =(0.001 - 0.00001825) radian. VII, § 58] TRIGONOMETRIC RELATIONS 77 The mil is therefore approximately one thousandth of a radian. (Hence its name.)* Since (§57) length of arc = radius x central angle in radians, it follows that we have approximately length of arc = x central angle in mils ; 1000 i.e. length of arc in yards a (radius in thousands of yards) • (angle in mils). The error here is about 2 % . Example 1. A battery occupies a front of 60 yd. If it is at 5500 yd. range, what angle does it subtend (Fig. 47)? We have, evidently, angle = —= 11 mils. 5.5 Example 2. Indirect Fire, t A battery posted with its right gun at G is to open fire on a battery at a point T, distant 2000 yd. and in- visible from G (Fig. 48). The officer directing tfie fire takes post at a point B from which both the target T and a church spire P, distant 3000 yd. from <?, are visible. B is 100 yd. at the right of the line 6? T and 120 yd. at the right of the line GP and the officer finds by measurement that the angle PBT contains 3145 mils. In order to train the gun on the P target the gunner must set off the angle PG T on the sight of the piece and then move the gun Fig. 48 * To give an idea of the value in mils of certain angles the following has been taken from the Drill Regulations for Field Artillery (1911), p. 164: " Hold the hand vertically, palm outward, arm fully extended to the front. Then the angle subtended by the width of thumb is 40 mils width of first finger at second joint is . ; . . .40 mils width of second finger at second joint is .... 40 mils width of third finger at second joint is 35 mils width of little finger at second joint is 30 mils width of first, second, and third fingers at second joint is . 115 mils These are average values." , t The limits of the text preclude giving more than a single illustration of the problems arising in artillery practice. For other problems the student is referred to the Drill Regulations for Field Artillery (1911) , pp. 57, 61, 150-164 ; and to Andrews, Fundamentals of Military Service, pp. 153-159, from which latter text the above example is taken. 78 PLANE TRIGONOMETRY [VII, § 58 until the spire P is visible through the sight. When this is effected, the gun is aimed at T. Let F and E be the feet of the perpendiculars from B to GT and GP respectively, and let B T' and BP' be the parallels to G T and GP that pass through B. Then, evidently, if the officer at B measures the angle PBT, which would be used instead of angle PG T were the gun at B in- stead of at G, and determines the angles TBT' = FTB and PBP 1 = EPB, he can find the angle PG T from the relation PGT = PBT = PBT- TBV-PBP*. Now tan FTB = — , tan EPB = — . TF PE small compared with G T and GP respectively, the radian measure of the angle is approximately equal to the tangent of the angle. Why ? Hence we have FB) FTB = tan FTB GT EPB = tan EPB = — GP approximately. Therefore TBT' = FTB = — radians = 50 mils, 2000 PBP 1 = EPB = i^- radians = 40 mils. 3000 Hence PGT = PBT - TBT' - PBP 1 = 3145 - 50 - 40 = 3055 mils, which is the angle to be set off on the sight of the gun. Hence from the situation indicated in Fig. 48 we have the following rule : (1) Measure in mils the angle PBT from the aiming point P to the target T as seen at B. (2) Measure or estimate the offsets FB and EB in yards, the range G T and the distance GP of the aiming point P in thousands of yards. (3) Compute in mils the offset angles by means of the relations TBT' = FTB, PBP' = EPB, TBT' = ^ B ~- GT PBP' = — • GP (4) Then the angle of deflection PGT is equal to the angle PBT diminished by the sum of the offset angles. VII, § 59] TRIGONOMETRIC RELATIONS 79 EXERCISES 1. A battery occupies a front of 80 yd. It is at 5000 yd. range. What angle does it subtend ? 2. In Fig. 48 suppose PBT = 3000 mils, FB = 200 yd., G T = 3000 yd., EB = 150 yd., GP = 4000 yd. Find the number of mils in PG T. 3. A battery at a point G is ordered to take a masked position and be ready to fire on an indicated hostile battery at a point T whose range is known to be 2100 yd. The battery commander finds an observing station B, 200 yd. at the right and on the prolongation of the battery front, and 175 yd. at the right of PG. An aiming point P, 5900 yd. in the rear, is found, and PBT is found to be 2600 mils. Find PG T. 4. A battery at a point G is to fire on an invisible object at a point T whose range is known to be 2000 yd. A battery commander finds an observing station B, 100 yd. at the right of G T and 150 yd. at the right of GP. The aiming point P is 1500 yd. in front and to the left of G T. The angle TBP contains 1200 mils. Find PG T. 59. The Sine Function. Let us trace in a general way the variation of the function sin 6 as 6 increases from 0° to 360°. For this purpose it will be convenient to think of the distance r as constant, from which it follows that the locus of P is a circle. When 6 = 0°, the point P lies on the #-axis and hence the ordinate is 0, i.e. sin 0° = 0/r = 0. As 6 increases to 90°, the ordinate increases until 90° is reached, when it becomes equal to r. Therefore, sin 90° = r/r = 1. As FlG 49 increases from 90° to 180°, the ordinate de- creases until 180° is reached, when it becomes 0. Therefore sin 180° = 0/r = As $ increases from 180° to 270°, the ordi- nate of P continually decreases algebraically and reaches its smallest algebraic value when = 270°. In this position the ordinate is — r and sin 270° = — r/r = — 1. When enters the fourth quadrant, the ordinate of P increases (algebraically) until the angle reaches 360°, when the ordinate becomes 0. 80 PLANE TRIGONOMETRY [VII, § 59 Hence, sin 360° = 0. It then appears that : as 6 increases from 0° to 90°, sin increases from to 1 ; as increases from 90° to 180°, sin 6 decreases from 1 to ; as increases from 180° to 270°, sin decreases from to — 1 ; as 6 increases from 270° to 360°, sin 6 increases from — 1 to 0. It is evident that the function sin 6 repeats its values in the same order no matter how many times the point P moves around the circle. We express this fact by saying that the function sin 6 is periodic and has a period of 360°. In symbols this is expressed by the equation sin [8 + n • 360°] = sin 9, where « is any positive or negative integer. The variation of the function sin 6 is well shown by its graph. To construct this graph proceed as follows : Take a system of rectangular axes and construct a circle of unit radius Fig. 50 with its center on the #-axis (Fig. 50). Let angle XM 4 P = 0. Then the values of sin 6 for certain values of 6 are shown in the unit circle as the ordinates of the end of the radius drawn at an angle 6. e 30° 45° 60° 90° sin MiP t M t P t M Z P Z M 4 P 4 ... Now let the number of degrees in be represented by dis- tances measured along OX. At a distance that represents 30° erect a perpendicular equal in length to sin 30° ; at a distance VII, § 60] TRIGONOMETRIC RELATIONS 81 that represents 60° erect one equal in length to sin 60°, etc. Through the points 0, P l9 P 2 , — draw a smooth curve ; this curve is the graph of the function sin 0. If from any point P on this graph a perpendicular PQ is drawn to the ic-axis, then QP represents the sine of the angle represented by the segment OQ. Since the function is periodic, the complete graph extends indefinitely in both directions from the origin (Fig. 51). 1&*X ilar to those 60. The Cosine Function. By arguments s used in the case of the sine function we may show that : as 8 increases from 0° to 90°, the cos 6 decreases from 1 to ; as increases from 90° to 180°, the cos decreases from to — 1 ; as 6 increases from 180° to 270°, the cos increases from — 1 to ; as 6 increases from 270° to 360°, the cos increases from to 1. The graph of the function is readily constructed by a method Fig. 52 similar to that used in the case of the sine function. This is illustrated in Fig. 52. The complete graph of the cosine function, like that of the sine function, will extend indefinitely from the origin in both 82 PLANE TRIGONOMETRY [VII, § 60 directions (Fig. 53). Moreover cos 6, like sin 6, is periodic and has a period of 360°, i.e. COS [6 4- 71 • 360°] as cos 6, where n is any positive or negative integer. Y 61. The Tangent Function. In order to trace the varia- tion of the tangent function, consider a circle of unit radius with^its center at the origin of a system of rectangular axes (Fig. 54). Then construct the tangent to this circle at the point M(l, 0) and let P denote any point on this tangent line. If angle MOP = 0, we have tan 6 = MP/OM ae MP/1 = MP, i.e. the line MP represents tan0. Now when $ = 0°, MP is 0, i.e. tan 0° is 0. As the angle 6 increases, tan 6 increases. As approaches 90° as a limit, MP becomes infinite, i.e. tan 6 becomes larger than any number whatever. At 90° the tangent is undefined. It is sometimes convenient to express this fact by writing tan 90° =oo. However we must remember that this is not a definition for tan 90°, for oo is not a number. This is merely a short way of saying that as approaches 90° tan becomes infinite and that at 90° tan is undefined. Thus far we have assumed to be an acute angle approach- ing 90° as a limit. Now let us start with as an obtuse angle Fig. 54 VII, § 61] TRIGONOMETRIC RELATIONS 83 and let it decrease towards 90° as a limit. In Fig. 55 the line MP' (which is here negative in direction) represents tan 0. Arguing precisely as we did before, it is seen that as the angle approaches 90° as a limit, tan 6 again increases in magni- tude beyond all bounds, i.e. becomes infi- nite, remaining, however, always negative. We then have the following results. (1) When is acute and increases to- wards 90° as a limit, tan always remains positive but becomes infinite. At 90° tan is undefined. (2) When is obtuse and decreases towards 90° as a limit, tan 6 always remains negative but becomes infinite. At 90° tan 6 is undefined. It is left as an exercise to finish tracing the variation of the tangent function as 6 varies from 90° to 360°. Note that tan 270°, like tan 90°, is undefined. In fact tan n • 90° is unde- fined, if n is any odd integer. Fig. Fig. 56 To construct the graph of the function tan 6 we proceed along lines similar to those used in constructing the graph of sin 6 and cos 0. The following table together with Fig. 56 illustrates the method. 84 PLANE TRIGONOMETRY [VII, § 61 e 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° tan MP X MP 2 MP Z undefined MP A MP b MP 6 ilfP 7 =0 MP X It is important to notice that tan 0, like sin 6 and cos 0, is periodic, but its period is 180°. That is tan(e + n-180 o )=tan6, where w is any positive or negative integer. X EXERCISES 1. What is meant by the period of a trigonometric function ? 2. What is the period of sin ? cos ? tan ? 3. Is sin defined for all angles ? cos ? 4. Explain why tan is undefined for certain angles. Name four angles for which it is undefined. Are there any others ? 5. Is sin (0 + 360°) = sin ? 6. Is sin (0 + 180°) = sin ? 7. Is tan ( + 180°) = tan ? 8. Is tan (0 + 360°) = tan'0 ? Draw the graphs of the following functions and explain how from the graph you can tell the period of the function : 9. sin0. 11. tan0. 13. sec0. 10. cos0. 12. csc0. 14.' ctn0. Verify the following statements : 15. sin90° + sin270° = 0. 18. cos 180° + sin 180° =- 1. 16. cos 90° + sin0° = 0. 19. tan 360° + cos 360° = 1. tan 1 80° + cos 1 80° = - 1 . 20 . cos 90° + tan 180°- I sin270^ = 1.^ 21. Draw the graphs of the functions sin 0, cos 0, tan 0, making use of a table of natural functions. See p. 112. \2fc) Draw the curves y = 2 sin ; y = 2 cos ; y = 2 tan 6. 23. Draw the curve y = sin + cos 0. 24. From the graphs determine values of for which sin = \ ; sin = 1 ; tan = 1; cos = \ ; cos = 1. VII, § 63] TRIGONOMETRIC RELATIONS 85 62. The Trigonometric Functions of — 9. Draw the angles 6 and — 0, where OP is the terminal line of and OP is the terminal line of — 6. Figure 57 shows an angle 6 in each of r Fig 57 the four quadrants. We shall choose OP = OP and («, y) as the coordinates of P and (x', y') as the coordinates of P'. In all four figures t! =» x, y' = - y, r' = r. Hence sin(-0) = ^ = :^ = -sin0, r r cos ( — 6) m — == - = cos 6, r' r _?/ y — tan ( - 0) = 2- = —a = - tan (9. Also, esc ( — 6) = — esc 6 ; sec ( — 0) = sec ; ctn ( — 6) = — ctn 0. The above results can be stated as follows : The functions of — 6 equal numerically the like named functions of 6. The algebraic sign, however, will be opposite except for the cosine and secant. Example, sin- 10° = -sin 10°, cos- 10° = cos 10°, tan-10°= -tan 10°. 63. The Trigonometric Functions of 180° + 6. Similarly, the following relations hold : sin (180° + 0) = — sin 0, esc (180° + 6) = - esc 0, cos (180° + 6) = - cos 0, sec (180° + 6) = - sec 6, tan (180° + 6) = tan 0, ctn (180° + 6) = ctn 0. The proof is left as an exercise. 86/ PLANE TRIGONOMETRY [VII, § 64 64. Summary. An inspection of the results of §§ 27-28, 62-63 shows : 1. Each f miction of — or 180° ± is equal in absolute value (but not always in sign) to the same function of 0. 2. Each function of 90° — is equal in magnitude and in sign to the corresponding co-function of 6. These principles enable us to find the value of any function of any angle in terms of a function of a positive acute angle (not greater than 45° if desired) as the following examples show. Example 1. Reduce cos 200° to a function of an angle less than 45°. Since 200° is in the third quadrant, cos 200° is negative. Hence cos 200° = - cos 20°. Why ? Example 2. Reduce tan 260° to a function of an angle less than 45°. Since 260° is in the third quadrant, tan 260° is positive. Hence tan 260° = tan 80° = ctn 10° (§ 27). Example 3. Reduce sin (— 210°) to a function of a positive angle less than 45°. From § 62 we know sin — 210° = — sin 210°. Considering the positive angle 210°, we have sin - 210° = - sin 210° = - [ - sin 30°] = sin 30°. EXERCISES Reduce to a function of an angle not greater than 45° : 1. sin 163°. 5. esc 901°. 2. cos (-110°). *"+ i. ctn (-1215°). + | Ans. -sin 20°. 7> tan 840°. -> 3. sec (-265°). 8. sin 510°. 4. tan 428°. tX— tv. Eind without the use of tables the values of the following functions : — >9. cos 570°. 11. tan 390°. 13. cos 150°. 10.' sin 330°. ^* 12. sin 420°. 14. tan 300°. Reduce the following to functions of positive acute angles : ^15. sin 250°. T* 18. sec (-245°). Ans. — sin 70° or — cos 20°. 19. C sc(— 321°). 16. cos 158°. 20. sin 269°. 17. tan (-389°). VII, § 64] TRIGONOMETRIC RELATIONS 87 Prove the following relations from a figure : (a) sin (90° + 0) = cos 0. (O sin (180° + 0) = — sini cos (90° + 0) = — sin 0. cos (180°+ 0) = -cos tan (90° + 0) = — ctn 0. tan (180° + 0)= tan0. esc (90° + 0)=sec0. csc(18O° + 0) = — csci sec (90° + 0) = - esc 0. sec (180° + 0) = -seci ctn (90° + 0) = - tan 0. ctn (180° + 0)=ctn0. (b) sin (180°- 6)=sm0. (<*) sin (270° -0) =-cos cos (180° — 6) = — cos 6. cos (270° —0) = - sin i tan (180° - 0) = -tan0. tan (270° - 0) = ctn 0. esc (180° — d) = esc 6. esc (270° — 0) = — sec sec (180° — d) = — sec0. sec (270° - 0) = - esc ctn (180° -0) = - ctn 6. ctn (270° - 0) = tan 0. (e) sin (270° + 0) = - cos 0. cos (270° + 0) = sin 0. tan (270° + 0) = - ctn 0. esc (270° + 0) = - sec 0. sec (270° + 0) = esc 0. ctn (270° + 0) = - tan 0. t J 4t^r Hn ik a o B e ta csa — fcuk MlljJIlMllH CHAPTER VIII TRIGONOMETRIC RELATIONS (Continued) ^5. Trigonometric Equations. An identity, as we have seen (§ 26), is an equality between two expressions which is satisfied for all values of the variables for which both expres- sions are defined. If the equality is not satisfied for all values of the variables for which each side is defined, it is called a conditional equality, or simply an equation. Thus 1 — cos = is true only if = n • 360°, where n is an integer. To solve a trigonometric equation, i.e. to find the values of for which the equality is true, we usually proceed as follows. 1. Express all the trigonometric functions involved in terms of one trigonometric function of the same angle. 2. Find the value (or values) of this function by ordinary algebraic methods. 3. Eind the angles between 0° and 360° which correspond to the values found. These angles are called particular solutions. 4. Give the general solution by adding n • 360°, where n is any integer, to the particular solutions. Example 1. Find 6 when sin 6 = $. The particular solutions are 30° and 150°. 30° + n ■ 360°, 150° + n • 360°. The general solutions are Example 2. Solve the equation tan 6 sin d — sin = 0. Factoring the expression, we have sin (tan 6 — Y)= 0. Hence we have sin = 0, or tan 6 — 1 = 0. Why ? The particular solutions are therefore 0°, 180°, 45°, 225°. The genera! solutions are n . 360°, 180° + n . 360°, 45° + n • 360°, 225° + n • 360°. 88 2. sin = — — • 2 , 3. 2 4. 2 5. tan0 = — 1. JL ctn 0=1. 16. 2 sin = tan 0. VIII, § 66] TRIGONOMETRIC RELATIONS 89 EXERCISES Give the particular and the general solutions of the following equations : tJq 7. sec — 2. 2 8. tan = 0. Vi 9. sec 2 = 2. 10. sin 2 = |. 11. cos0= — £. 12. csc 2 = f /l3. 4 sin — 3 esc = 0. 1 14. 2 sin cos 2 = sin 0. )l5. cos -f sec = f . ^Irw. Particular solutions : 0°, 180°, 60°, 300°. 17. 3 sin + 2 cos = 2. /l8. 2 cos 2 0—1 = 1 — sin 2 0. Inverse Trigonometric Functions. The equation x — sin y (1) may be read : y is an angle whose sine is equal to x, a statement which is usually written in the contracted form y = arc sin x.* (2) For example, x = sin 30° means that x = \, while y = arc sin i means that y = 30°, 150°, or in general (n being an integer), 30° + n • 360° ; 150° + n • 360°. Since the sine is never greater than 1 and never less than — 1, it follows that —l_\x—\l. It is evident that there is an unlimited number of values ofy = arc sin x for a given value of x in this interval. We shall now define the principal value Arc sin x f of arc sin x, distinguished from arc sin x by the use of the capital A, to be * Sometimes written y = sin -1 x. Here — 1 is not an algebraic exponent, but merely a part of a functional symbol. When we wish to raise sin x to the power — 1, we write (sin x)-}. t Sometimes written Sin-i x, distinguished from sin -1 x by the use of the capital S. 90 PLANE TRIGONOMETRY [VIII, § 66 the numerically smallest angle whose sine is equal to x. This func- tion like arc sin x is denned only for those values of x for which The difference between arc sin x and Arc sin x is well illus- trated by means of their graph. It is evident that the graph ofy = arc sin x, i.e. x = sin y is simply the sine curve with the role of the x and y axes inter- changed. (See Fig. 58.) Then for every admissible value of x, there is an un- limited number of values of y ; namely, the ordinates of all the points P 1} P 2 , •-, in which a line at a distance x and parallel to the 2/-axis intersects the curve. The single-valued function Arc sin x is repre- sented by the part of the graph between M and N, Similarly arc cos x, defined as " an angle whose cosine is x," has an unlimited' number of values for every admissible value of x(— 1 f^ x < 1) We shall define the principal value Arc cos x as the smallest positive angle whose cosine is x. That is. Fig. 58 ^ Arc cos x <^ 7r. Figure 59 represents the graph of y = arc cos x, and the portion of this graph between M and N represents Arc cos x. Similarly we write x = tan y as y = arc tan x, and in the same way we define the symbols arc ctn x ; arc sec x ; arc esc x. The principal values of all the inverse trigonometric functions are given in the following table. Y 2tt 37T P» N \ IT 7T 2 ft M -1 Pi 1 X V= arc cos x y=Arc cos x Fig. 59 VIII, § 66] TRIGONOMETRIC RELATIONS 91 y- Arc sin x Arc cos x Arc tan x Range of x -lgx^ 1 -l^a^l all real values Range of y 7T . 7T to — 2 2 tO 7T to — 2 2 x positive 1st Quad. 1st Quad. 1st Quad. x negative 4th Quad. 2d Quad. 4th Quad. Arc ctn x Arc sec x Arc cscx Range of x all values x^l orx^- 1 a;^lorx2-l Range of y OtOT tO 7T to — 2 2 x positive 1st Quad. 1st Quad. 1st Quad. x negative 2d Quad. 2d Quad. 4th Quad. In so far as is possible we select the principal value of each inverse function, and its range, so that the function is single- valued, continuous, and takes on all possible values. This ob- viously cannot be done for the Arc sec x and for Arc esc y. EXERCISES 1. Explain the difference between arc sin x and Arc sin x. >^. Find the values of the following expressions : \(a) Arc sin \. (d) Arc tan — 1. (e) arc cos V3 (/) Arc cos 22. >>^&) arc sin \. (c) arc tan 1. 2 S^What is meant by the angle it ? tt/4 ? 4. Through how many radians does the minute hand of a watch turn in 30, minutes ? in one hour ? in one and one half hours ? 6. For what values of x are the following functions defined : y\d) arc sin x ? ^/($) arc tan x ? _--^ e ) arc sec x ? (6) arc cos x ? (d) arc ctn x ? (/) arc esc x ? 6. What is the range of values of the functions : (or) Arc. sin x ? (c) Arc tan x ? (e) Arc sec x. (6) Arc cos x ? (d) Arc ctn x ? (/) Arc esc x ? fa I <0 TW 92 PLANE TRIGONOMETRY [VIII, § 66 7. Draw the graph of the functions : (a) arc sin x. (c) arc tan x. (e) arc sec x. (&-) arc cos x. (d) arc etna;. (/) arc esc x. 8. Find the value of cos (Arc tan f). Hint. Let Arc tan f = 6. Then tan d = £ and we wish to find the value of cos 6. : 9. Find the values of cos (arc tan f ) . !^V> • TIC Find the value of the following expressions : (a) sin (arc cos |). *"* -(c) cos (Arc cos T 5 ^). (e) sin (Arc sin \). (6) sin (arc sec 3). {$) sec (Arc esc 2). (/) tan (Arc tan 5) . 11. Prove that Arc sin (2/5)= Arc tan (2/V21) 12. Find x when Arc cos (2 x 2 - 2 x) = 2 tP/3. \ ^- Find the values of the following expressions : 13. cos [90°— Arc tan f]. j£f 1^1 f 14. sec [90° — Arc sec 2].- 15. tan [90° - Arc sin T \]. 67. Projection. Consider two directed lines p and q in a plane, i.e. two lines on each of which, one of the directions has been specified as positive (Fig. 60). Let A and B be any two points on p and let A', B' be the points in which per- Fig. 00 pendiculars to q through A and B, respectively, meet q. The directed segment A'B' is called the projection of the directed seg- ment AB on q and is denoted by A'B' = proj ff AB. In both figures AB is positive. In the first figure A'B' is posi tive, while in the second figure it is negative. As special cases of this definition we note the following : VIII, § 67] TRIGONOMETRIC RELATIONS 93 1. If p and q are parallel and are directed in the same way, we have proj, AB = AB. 2. If p and q are parallel and are directed oppositely, we have proj ff AB = — AB. 3. If p is perpendicular to q, we have proj, AB = 0. It should be noted carefully that these propositions arc true no matter how A and B are situated on p. We may now prove the following important proposition : If A and B are any two points on a directed line p, and q is any directed line in the same plane with p, then we have both in magnitude and sign (1) projg AB = AB ■ cos (pq)* = AB . cos (qp). We note first from § 8 that (pq)+ (qp) = + n- 360°, where n is any integer. Hence from § 64, cos (jxj) = cos (qp). Two cases arise. Jt T22 Fig. 61 Case 1. Suppose AB is positive, i.e. it has the same direc- tion as p. Through A draw a line q^ parallel to q and with the same direction. [It is evident that we may assume without loss of generality that q is horizontal and is directed to the right.] Let A'B' be the projection of AB on q and let BB' meet q x in B x . Then by the definition of the cosine we have AB ——± = cos (qip) = cos (pqi) = cos (qp) = cos ( pq) AB * (pq) represents an angle through which p may be rotated in order to make its direction coincide with the direction of q ; similarly for (qp). 94 PLANE TRIGONOMETRY [VIII, § 67- in magnitude and sign. Hence AB± = AB ■ cos (pq) = AB • cos (qp). But AB X = A'B' = proj 3 AB. Therefore proj tf AB = AB • cos (pp) = AB • cos (qp). Case 2. Suppose AB is negative. If AB is negative, BA is positive and we have from Case 1, B'A! = BA • cos (pq) = BA • cos (qp). Changing the signs of both members of this equation, we have A'B' = AB • cos (})q)= AB • cos (qp). The special cases 1, 2, 3, are obtained from formula (1) by placing (qp) or (pq) equal to 0°, 180°, 90° respectively. Theorem. If A, B, C are any three points in a plane, and I is any directed line in the plane, the algebraic sum of the projec- tions of the segments AB and- BC on I is equal to the projection of the segment AC on I. As a point traces out the path from A to B, and then from B to C (Fig. 62), the projection of the point traces out the segments from A' to B' and then from B' to C. The tjjjft result of this motion is a motion from A' to O which represents the projection of AC, i.e. A'B' + B'C = A' C. I / — ' — ■ z^ B ti C >^ A' ( i' h ' EXERCISES 1. What is the projection of a line segment upon a line I, if the line segment is perpendicular to the line I ? 2. Find proj x ^4JB and proj^l?* in each of the following cases, if a denotes the angle from the x-axis to AB. (a) AB = 5, a = 60°. (c) AB = 6, a = 90°. (6)^45 = 10, a = 300°. (d) AB = 20, a = 210°. * Proj x AB and proj,, AB mean the projections of AB on the x-axis and the y-axis, respectively. <^ w- <-^ VIII, § 68] TRIGONOMETRIC RELATIONS 95 3. Prove by means of projection that in a triangle ABC a—b cos C -f c cos B. 4. If projj. AB = 3 and proj„ AB = —4, find the length of AB. 5. A steamer is going northeast 20 miles per hour. Hots fast is it going north ? going east ? 6. A 20 lb. block is sliding down a 15° incline. Find what force acting directly up the plane will just hold the block, allowing ope half a pound for friction. 7. Prove that if the sides of a polygon are projected in order upon any given line, the sum of these projections is zero. Fig. 63 The Addition Formulas. We may now derive formulas for sin (a -f- /3), cos (a -f- ft), and tan (a + ft) in terms of func- tions of a and ft. To this end let P(x, y) be any point on the terminal side of the angle a (the initial side being along the posi- tive end of the a>axis and the vertex being at the origin). The angle a + ft is then obtained by rotating OP through an an^le ft. If P' (x', y') is the new Sta- tion P after this rotation and OP = OP' = r, we have sin (a -f- ft) = £ , cos (a + ft) = - , by v r definition. Our first problem is, therefore, to find x' and y' in terms of x, y, and ft. In the figure OMP is the new position of the triangle OMP after rotating it about through the angle ft. Now, x' = proj x OP' ss proj x OM' + proj x M'F = xcosft + ycos(ft + ^\ = x cos ft — y sin ft. 96 PLANE TRIGONOMETRY [VIII, § 68 Similarly, y> = proj, OP' = proj, OM' + proj tf M'F = x cos(?-- ft\+ y cos ft = x sin ft + y cos /?. Hence, , , ~ x y' x ,-+ n . V n ' sm (« + j3)=V-=- sr$/3+^ cos £ r r = sin a cojr [ or (1) m sin (a + P) = sin a co# p -f cos a sin p. Also cos s(« + £) = ^- = -^osft-^ sin/?. or (2) cos (a + p) = cos a cos p — sin a sin p. Further we have tan (a 4- B) = S * n ( a ~*~ ® = sni g cos ft + cos <* sin ft cos (a 4- ft) cos a cos ft — sin a sin /J Dividing numerator and denominator by cos a cos ft, we have (3) • tan(a+B)= tan * + tan P. w v K; 1 - tan a tan p Furthermore, by replacing ft by — ft in (1), (2), and (3), and recalling that sin (— ft) = — sin ft, cos (— ft) = cos ft, tan (— ft) = — tan ft, we obtain -^^fc_ (4) sin (a — P) = sin a cmf$ — cos a sin p, (5) cos (a — P) = cos a cm$ ■+- sin a sin p, (6) tan (a. - tt = tan o^ tan p tan (a - p) = — y — r v r/ 1 + t*n a tan p EXERCISES Expand the iollowing : -*-±r sin (45° + «) = 3. cos (60° + a) = 5. sin (30° - 45°) = —=«. tan (30° - 0) = 4. tan (45° + 60°) = . 6. cos (180° - 45°) = 7. What do the following formulas become if « = /3 ? sin (« + (S)= sin a cos /3 + cos a sin p. t (a A- 8 s )— tan a + tan P . sin (a — /3) = sin a cos /3 — cos a sin 0. * 1 — tan a tan /3 cos (a + /3) = cos a cos — sin a sin /S. . , _ q\ _ tan a — tan g ( cos (a — /3) = cos a cos p + sin a sin j8. 1 + tan a tan /3 VIII, § 68] TRIGONOMETRIC RELATIONS 97 8. Complete the following formulas : sin 2 a cos a + cos 2 a sin a — tan 2 a + tan a _ sin 3 a cos a — cos 3 a sin a = 1 — tan 2 a tan a -^* Prove sin 75° = V ^ + 1 , cos 75° = V ^ ~ 1 , tan75° = V g + 1 - 2V2 2V2 V3-1 10. Given tan a = f , sin ft = T 5 ^, and a and ft both positive acute angles, find the value of tan (a + ft); shr(a~— ft); cos (a + ft); tan (a — ft). ,-. »r1 1. Prove that (a) cos (60° + a) + sin (30° + a) = cos a. * ( 6) sin (60° + 0) - sin (60° - 6) - sin 0. (c) cos (30° + 0)- cos (30° - 0)= - sin 6. (d) cos (45° + 6) + cos (45° - 0) = V2 • cos 0. > (e) sin 1 a + - ) + sin ( a — — j = sin a. (/) cos ( a + - ) + cos (a — -) = V3 • cos a. ~* — 12. By using the functions of 60° and 30° find the value of sin 90° ; cos 90°. 13. Find in radical form the value of sin 15° ; cos 15° ; tan 15° ; sin 105° ; cos 105° ; tan 105°. 14. If tan a = |, sin ft = T 5 T , and a is in the third quadrant while ft is in the second, find sin (a ± ft) ; cos (a ± ft) ; tan (a ± ft). Prove the following identities : <^"~" 15 sin (a + ft) _ tan a + tan ft _ 16 sin 2 a , cos 2 a _ sm 3 a sin (a — ft) tana — tan ft sec a esc a 17 tana -tan (a -ft) = tan ^ 19. ( a ) sin ( 180 o _ 9) - s i n $m 1 + tan a tan (a— ft) (6) cos (180° - 6) = - cos 0. x 18. tan(0±45°) + ctn(0T45°)=O. (c) tan (180° - 6) = - tan 0. 20. cos (a -f ft) cos (a — ft) = cos 2 a — sin 2 ft. 21. sin (a + ft) sin (a — ft) = sin 2 a - sin 2 ft. 22. ctn(« + /9) = ctnttctn g- 1 . 23. ctn (a - ft) = Ctn " ctn ** + * . ctna + ctnft ctnft — etna 24. Prove Arc tan £ + Arc tan | = ?r/4. [Hint : Let Arc tan \ = x and Arc tan \ = y. Then we wish to prove x + y = ir/4, which is true since tan (x + y)= 1.] 25. Prove Arc sin a + Arc cos a = - if < a < 1. p 26. Prove Arc sin T * 7 -f Arc sin | = Arc sin ||. i H A 98 x PLANE TRIGONOMETRY [VIII, § 68 27. Prove Arc tan 2 + Arc tan £ as ir/2. 28. Prove Arc cos § + Arc cos (— T 5 y) = Arc cos (— f|). 29. Prove Arc tan T 8 5 + Arc tan f = Arc tan f £. -30. Find the value of sin [Arc sin | + Arc ctnf ]. - 31. Find the value of sin [Arc sin a + Arc sin 6] if < a < 1, < b < 1. 32. Expand sin (x + y + z) ; cos(x + y + z). [Hint : x + y + z =(x + y)+ z.] 33. The area i of a triangle was computed from the formula A = I ab sin 0. If an error c was made in measuring the angle 0, show that the corrected area A' is given by the relation.^.' = A(cos e + sin e ctn 6). 69. Functions of Double Angles. In this and the follow- ing articles (§§ 69-71) we shall derive from the addition formulas a variety of other relations which are serviceable in transforming trigonometric expressions. Since the formulas for sin (a + fi) and cos (a + /?) are true for all angles a and (3, they will be true when /? = a. Putting /3 = a, we obtain (1) sin 2 a = 2 sin a cos a, (2) cos 2 a = cos 2 a — sin 2 a. Since sin 2 a + cos 2 a — 1, we have also (3) cos 2 a = 1 - 2 sin 2 a (4) =2cos 2 a-l. Similarly the formula for tan (a + ft) (which is true for all angles a, ft, and a+ft which have tangents) becomes, when ft=a, (5) tan2q= 2tana , v ; l-tan 2 a which holds for every angle for which both members are denned. The above formulas should be learned in words. For ex- ample, formula (1) states that the sine of any angle equals twice the sine of half the angle times the cosine of half the angle. Thus sin6^ = 2 sin3« cos3^, 2 tan 2 x tan 4 x — l-tan 2 2x' cos x = cos 2 - — sin 2 -> VIII, § 70] TRIGONOMETRIC RELATIONS 99 70. Functions of Half Angles. From (3), § 69, we have Therefore 2sin 2 £=l — cos a. (6) «in«_ | /I -cos a ° m 2 _± \ 2 From (4), § 69, we have 2 cos 2 - = 1 -+- cos a. Therefore (7) cos« = ± /±f5^. Formulas (6) and (7). are at once seen to ! «. Now, if we divide formula (6) by formula (7), we obtain /QX * a /l — cos a (8) tan - = ± \/- , v } 2 ' V'l + cosa' which is true for all angles a except n • 180°, where n is any- odd integer. Example. Given sin^. =— 3/5, cos 4 negative ; find sin (A/2). Since the angle A is in the third quadrant, A/2 is in the second or fourth quadrant, and hence sin (A/2) may be either positive or negative. Therefore, since cos A = — 4/5, we have 2 \ 2 «/Tn 10 VTo io EXERCISES Complete the following formulas and state whether they are true for all angles : 1. sin 2 a = 3 - tan 2 a — 5. cos " = A 2. cos2a= (three forms). 4 s in-= 6. tan - = 2 2 7. In what quadrant is 0/2 if 6 is positive, less than 360°, and in the second quadrant ? third quadrant ? fourth quadrant ? 8. Express cos 2 a in terms of cos 4 a. 9. Express sin 6 x in terms of functions of 3 x. 100 PLANE TRIGONOMETRY [VIII, § 70 10. Express tan 4 a in terms of tan 2 a. 11. Express tan 4 a in terms of cos 8 a. 12. Express sin x in terms of functions of x/2. 13. Explain why the formulas for sin x and cos x in terms of functions of 2 x have a double sign. 14. From the functions of 30° find those of 60°. 15. From the functions of 60° find those of 30°. 16. From the functions of 30° find those of 15°. 17. From the functions of 15° find those of 7°. 5. 18. Find the functions of 2 a if sin a = $ and a is in the second quadrant. 19. Find the functions of a/2 if cos a =— 0.6 and a is in the third quadrant, positive, and less than 360°. 20. Express sin 3 a in terms of sin a. [Hint : 3a = 2a + a.] 21. From the value of cos 45° find the functions of 22°. 5. 22. Given sin a = — and a in the second quadrant. Find the values of (a) sin 2 a. (c) cos 2 a. (e) tan 2 a. (6) sin". (d) cos?. (/) tan|. 23. If tan 2 a = | find sin a, cos a, tan a if a is an angle in the third quadrant. Prove the following identities : 24. 1 + C0 *«=cto& 27. l-cos2fl + sin2fl =tan . sin a 2 . . 1 + cos 2 + sin 2 25. 26. Tsin — cos-] =1 — sin0. 28. sin- + cos — = ± Vl + sina. L.2 2J 22 cos2 + cos0 4-l „ ctn ,, j 29 Be0 a + tan«=ten^ + ^V sin20 + sin0 \4 2/ 30. 2 Arc cos a; = Arc cos (2 x 2 — 1). 31. 2 Arc cosx = Arc sin (2 xVl — x 2 ). 32. tan [2 Arc tanx] = ^^-. 34. tan [2 Arc sec x] = ± 2 ' 1 - x 2 J 2 - x 2 33. cos [2 Arc tan x] = — x • /35^ *os (2 Arc sin a) = 1 — 2 a 2 . 1 +x 2 Solve the following equations 36. cos 2 x + 5 sin x = 3. 40. sin 2 2 x — sin 2 x ss f . 37. cos2x — sinx = \. 41. sin2x = 2cosx. 38. sin 2 x cos x = sin x. 42. 2 sin 2 2 x = 1 — cos2x. 39. 2sin 2 x + sin 2 2x = 2. 43. ctnx — csc2x — 1. fa VIII, § 71] TRIGONOMETRIC RELATIONS 101 44. A flagpole 50 ft. high stands on a tower 49 ft. high. At what dis- tance from the foot of the tower will the flagpole and the tower subtend equal angles ? 45. The dial of a town clock h^s a diameter (j|f JO ft. and its center is 100 ft. above the ground. At 1 what' distance from the foot of the tower will the dial be must be as large most plainly viqbie f] £r he ajn^'fubWr-ded by the dial as possible.]* ° ' •••• ' 71. Product Formulas. From § 68 we have sin (a-\- (3) = sin a cos /? -f cos a sin /3, sin (a — /?) = sin a cos /? — cos a sin /?. Adding, we get (1) sin (a + p) + sin (a — /?) = 2 sin a cos /3. Subtracting, we have (2) sin (a + ft) — sin (« — p) = 2 cos a sin 0. Now, if we let a -f- /? = P and a — ft = Q, thell « = ^, = Z^$. Therefore formulas (1) and (2) become P -+- O P sin P + sin Q = 2 sin ^ v cos — 2 2 Pi Q p sin P — sin Q = 2 cos — — * sin — 2 Similarly, starting with cos (« + /?) and cos (a — /?) and per- forming the same operations, the following formulas result : P 4- O P — O cos P + cos Q = 2 cos — —-*- cos — —i-, A A cos P — cos Q = — 2 sin J~ v sin — —^. 2 2 y^. In words : the sum of two sines = twice sin (half sum) times cos (half difference), the difference of two sines = twice cos (half sum) times sin (half difference),* * The difference is taken, first angle minus the second. 102 PLANE TRIGONOMETRY [VIII, § 71 the sum of two cosines = twice cos (half sum) times cos (half difference), the difference of two cosines, = minus twice sin (half sum) times sin (half difference). * Example 1. -Prove that ' a f coB8a ; + co 8:a ?=ctn j a , sin 3 x + sin x for all angles for which both members are defined. cos 3 x + cos x _ 2 cos ^(3 x 4- x) cos |(3 x — x) _ cos 2 x _ . 9 sin 3 x + sin x 2 sin £(3 x + x) cos \ (3 x — x) ~" sin 2 x ~~ Example 2. Reduce sin 4 x 4- cos 2 x to the form of a product. We may write this as sin 4 x 4- sin (90° — 2x), which is equal to 2 sin Ix + W-Z* cos tx-W + az 2 sin (45 „ + x) cos (3 x _ 45 „ } _ EXERCISES Reduce to a product : 1. sin 4 — sin 2 0. 4. cos 2 + sin 2 0. 7. cos 3 x + sin 5 x. 2. cos + cos 3 0. 5. cos 3 — cos 6 0. 8. sin 20° — sin 60°. 3. cos 6 + cos 2 0. 6. sin (x -f Ax) — sin x. Show that 9. sin 20° + sin 40° = cos 10°. 12 s in 15° 4- sin 75° _ _ 6QO 10. cos 50° 4- cos 70° = cos 10°. ' sin 15° - sin 75° ~ 11. sin75 °- sinl5 ° = tan 30°. 13. sin3 0-sin5 = _ ^ 4 , cos 75° 4- cos 15° cos 3 — cos 5 Prove the following identities : , ^ " "^ sm~$TT4 r 'Slfi~3 ft _ g^fl" 15 sin a + sin ft _ tan \ (a + ft) cos 3 a — cos 4 a 2 ' sin a — sin ft tan £ (a — ft) . .. cos a 4- 2 cos 3 a 4- cos 5 a cos 3 a id. = • cos 3 a. + 2 cos 5 a 4- cos 7 a cos 5 a -_ cos a— cos ft _ _ tan ^(#4- ft) lg sin (n — 2) 4- sin nd _ . cos «4- cos ft ctn£(a — ft) ' cos (n — 2) — cos nd Solve the following equations : ■ 19. cos 4- cos 50 = cos 30. 22. sin 4 — sin 2 = cos 3 0. 20. sin 4- sin 5 = sin 3 0. 23. cos 7 — cos = — sin 4 21. sin 3 6 + sin 7 = sin 5 0. *The difference is taken, first angle minus the second. I . OW^& VIII. § 71] TRIGONOMETRIC RELATIONS 103 MISCELLANEOUS EXERCISES 1. Reduce to radians 65°, - 135°, - 300°, 20°. 2. Reduce to degrees 7r, 3 ?r, — 2 w, 4 v radians. 3. Find sin (a — /3) and cos (a + /S) when it is given that a and /3 are positiye and acute and tan a = f and sec /3 = *£. 4. Find tan (a + /S) and tan (a — /3) when it is given that tan a = \ and tan /S = |. 5. Prove that sin 4 a = 4 sin a cos a — 8 sin 3 a cos a. 6. Given sin = — - y and in the second quadrant. Find sin 2 V5 cos 2 0, tan 2 0. Prove the following identities : . 7 . sin2«= 2tan " ■ 9. sec2« csc2a 1 + tan 2 a esc 2 a — 2 8. cos2 ( , = 1 - tan2 ^. 10. tan«= sin2a 1 + tan 2 a 1 + cos 2 a s~ 11. sin (a + /S) cos /3 — cos (a + /3) sin = sin a. ^-£ll. sin 2 a + sin 2 j3 + sin 2 7 = 4 sin a sin sin 7, if a + /S + 7 = 180°. 1 + tan - c* , q. cos a 2 _d J <-~ ^ ' l-sta«"l_ta„!' "= r T\^P & — \_ ten -x. 4 ^_ 1 . A *• % <u*J~^&. ^ _ 1 4^»\A 5i_ .2- • *^-c4^rv^ U^ 6p Jf^iilO -*q sp OQ 0Ti + #f T# ^~ (ot +ft) ■* %*~oC(U->($ *6*->©^ u. « £_ ff ^^tu#^ m T<. u v ir.o m Ho' 5 R M* f * ' ^ r % * - To* 3 T? fi^L^ I /^/^C TABLES FOUR DECIMAL PLACES 106 [Moving the decimal poin Squares of Numbers t one place in N requires a corresponding move of two places in N 2 ] u N 2 1 2 3 4 5 6 7 8 9 0.0 .0000 .0001 .0004 .0009 .0016 .0025 .0036 .0049 .0064 .0081 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 .0100 .0400 .0900 .1600 .2500 .3600 .4900 .6400 .8100 .0121 .0441 .0961 .1681 .2601 .3721 .5041 .6561 .8281 .0144 .0484 .1024 .1764 .2704 .3844 .5184 .6724 .8464 .0169 .0529 .1089 .1849 .2809 .3969 .5329 .6889 .8649 .0196 .0576 .1156 .1936 .2916 .4096 .5476 .7056 .8836 .0225 .0625 .1225 .2025 .3025 .4225 .5625 .7225 .9025 .0256 .0676 .1296 .2116 .3136 .4356 .5776 .7396 .9216 .0289 .0729 .1369 .2209 .3249 .4489 .5929 .7569 .9409 .0324 .0784 .1444 .2304 .3364 .4624 .6084 .7744 .9604 .0361 .0841 .1521 .2401 .3481 .4761 .6241 .7921 .9801 1.0 1.000 1.020 1.040 1.061 1.082 1.103 1.124 1.145 1.166 1.188 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.210 1.440 1.690 1.960 2.250 2.560 2.890 3.240 3.610 1.232 1.464 1.716 1.988 2.280 2.592 2.924 3.276 3.648 1.254 1.488 1.742 2.016 2.310 2.624 2.958 3.312 3.686 1.277 1.513 1.769 2.045 2.341 2.657 2.993 3.349 3.725 1.300 1.538 1.796 2.074 2.372 2.690 3.028 3.386 3.764 1.323 1.563 1.823 2.103 2.403 2.723 3.063 3.423 3.803 1.346 1.588 1.850 2.132 2.434 2.756 3.098 3.460 3.842 1.369 1.613 1.877 2.161 2.465 2.789 3.133 3.497 3.881 1.392 1.638 1.904 2.190 2.496 2.822 3.168 3.534 3.920 1.416 1.664 1.932 2.220 2.528 2.856 3.204 3.572 3.960 2.0 4.000 4.040 4.080 4.121 4.162 4.203 4.244 4.285 4.326 4.368 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.410 4.840 5.290 5.760 6.250 6.760 7.290 7.840 8.410 4.452 4.884 5.336 5.808 6.300 6.812 7.344 7.896 8.468 4.494 4.928 5.382 5.856 6.350 6.864 7.398 7.952 8.526 4.537 4.973 5.429 5.905 6.401 6.917 7.453 8.009 8.585 4.580 5.018 5.476 5.954 6.452 6.970 7.508 8.066 8.644 4.623 5.063 5.523 6.003 6.503 7.023 7.563 8.123 8.703 4.666 5.108 5.570 6.052 6.554 7.076 7.618 8.180 8.762 4.709 5.153 5.617 6.101 6.605 7.129 7.573 8.237 8.821 4.652 5.198 5.664 6.150 6.656 7.182 7.728 8.294 8.880 4.796 5.244 5.712 6.200 6.708 7.236 7.784 8.352 8.940 9.000 9.060 9.120 9.181 9.242 9.303 9.364 9.425 9.486 9.548 9.610 10.24 10.89 11.56 12.25 12.96 13.69 14.44 15.21 9.672 10.30 10.96 11.63 12.32 13.03 13.76 14.52 15.29 9.734 10.39 11.02 11.70 12.39 13.10 13.84 14.59 15.37 9.797 10.43 11.09 11.76 12.46 13.18 13.91 14.70 15.44 9.860 10.50 11.16 11.83 12.53 13.25 13.99 14.75 15.52 9.923 10.56 11.22 11.90 12.60 13.32 14.06 14.82 15.60 9.986 10.63 11.29 11.97 12.67 13.40 14.14 14.90 15.68 10.05 10.69 11.36 12.04 12.74 13.47 14.21 14.98 15.76 10.11 10.76 11.42 12.11 12.82 13.54 14.29 15.05 15.84 10.18 10.82 11.49 12.18 12.89 13.62 14.26 15.13 15.92 4.0 16.00 16.08 16.16 16.24 16.32 16.40 16.48 16.56 16.65 16.73 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 16.81 17.64 18.49 19.36 20.25 21.16 22.09 23.04 24.01 16.89 17.72 18.58 19.45 20.34 21.25 22.18 23.14 24.11 16.97 17.81 18.66 19.54 20.43 21.34 22.28 23.23 24.21 17.06 17.89 18.65 19.62 20.52 21.44 22.37 23.33 24.30 17.14 17.98 18.84 19.71 20.61 21.53 22.47 23.43 24.40 17.22 18.06 18.92 19.80 20.70 21.62 22.56 23.52 24.50 17.31 18.15 19.01 19.89 20.79 21.72 22.66 23.62 24.60 17.39 18.23 19.10 19.98 20.88 21.81 22.75 23.72 24.70 17.47 18.32 19.18 20.07 20.98 21.90 22.85 23.81 24.80 17.56 18.40 19.27 20.16 21.07 22.00 22.94 23.91 24.90 5.0 25.00 25.10 25.20 25.30 25.40 25.50 25.60 25.70 25.81 25.91 Squares of Numbers 10' [Moving the decimal point one place in N requires a corresponding move of two places in N 2 ] I F o , . 3 4 5 6 7 8 9 5.0 25.00 25.10 j 25.20 25.30 25.40 25.50 25.60 25.70 25.81 25.91 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 26.01 27.04 28.09 29.16 30.25 31.36 32.49 33.64 34.81 26.11 27.14 28.20 29.27 30.36 31.47 32.60 33.76 34.93 26.21 27.25 28.30 29.38 30.47 31.58 32.72 33.87 35.05 26.32 27.35 28.41 29.48 30.58 31.70 32.83 33.99 35.16 26.42 27.46 28.52 29.59 30.69 31.81 32.95 34.11 35.28 26.52 27.56 28.62 29.70 30.80 31.92 33.06 34.22 35.40 26.63 27.67 28.73 29.81 30.91 32.04 33.18 34.34 35.52 26.73 27.77 28.84 29.92 31.02 32.15 33.29 34.46 35.64 26.83 27.88 28.94 30.03 31.14 32.26 33.41 34.57 35.76 26.94 27.98 29.05 30.14 31.25 32.38 33.52 34.69 35.88 6.0 36.00 36.12 36.24 36.36 36.48 36.60 36.72 36.84 36.97 37.09 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 37.21 38.44 39.69 40.96 42.25 43.56 44.89 46.24 47.61 37.33 38.56 39.82 41.09 42.38 43.69 45.02 46.38 47.75 37.45 38.69 39.94 41.22 42.51 43.82 45.16 46.51 47.89 37.58 38.81 40.07 41.34 42.64 43.96 45.29 46.65 48.02 37.70 38.94 40.20 41.47 42.77 44.09 45.42 46.79 48.16 37.82 39.06 40.32 41.60 42.90 44.22 45.56 46.92 48.30 37.95 39.19 40.45 41.73 43.03 44.36 45.70 47.06 48.44 38.07 39.31 40.58 41.86 43.16 44.49 45.83 47.20 48.58 38.19 39.44 40.70 41.99 43.30 44.62 45.97 47.33 48.72 38.32 39.56 40.83 42.12 43.43 44.76 46.10 47.47 48.72 7.0 49.00 49.14 49.28 49.42 49.56 49.70 49.84 49.98 50.13 50.27 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 50.41 51.84 53.29 54.76 56.25 57.76 59.29 60.84 62.41 50.55 51.98 53.44 54.91 56.40 57.91 59.44 61.00 62.57 50.69 52.13 53.58 55.06 56.55 58.06 59.60 61.15 62.73 50.84 52.27 53.73 55.20 56.70 58.22 59.75 61.31 62.88 50.98 52.42 53.88 55.35 56.85 58.37 59.91 61.47 63.04 51.12 52.56 54.02 55.50 57.00 58.52 60.06 61.62 63.20 51.27 52.71 54.17 55.65 57.15 58.68 60.22 61.78 63.36 51.41 52.85 54.32 55.80 57.30 58.83 60.37 61.94 63.52 51.55 53.00 54.46 55.95 57.46 58.98 60.53 62.09 63.68 51.70 £3.14 54.61 56.10 57.61 59.14 60.68 62.25 63.84 8.0 64.00 64.16 64.32 64.48 64.64 64.80 64.96 65.12 65.29 65.45 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 65.61 67.24 68.89 70.56 72.25 73.96 75.69 77.44 79.21 65.77 67.40 69.06 70.73 72.42 74.13 75.86 77.62 79.39 65.93 67.57 69.22 70.90 72.59 74.30 76.04 77.79 79.57 66.10 67.73 69.39 71.06 72.76 74.48 76.21 77.97 79.74 66.26 67.90 69.56 71.23 72.93 74.65 76.39 78.15 79.92 66.42 68.06 69.72 71.40 73.10 74.82 76.56 78.32 80.10 66.59 68.23 69.89 71.57 73.27 75.00 76.74 78.50 80.28 66.75 68.39 70.06 71.74 73.44 75.17 76.91 78.68 80.46 66.91 68.56 70.22 71.91 73.62 75.34 77.08 78.85 80.64 67.08 68.72 70.39 72.08 73.79 75.52 77.26 79.03 80.82 9.0 81.00 81.18 81.36 81.54 81.72 81.90 82.08 82.26 82.45 82.63 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 82.81 84.64 86.49 88.36 90.25 92.16 94.09 96.04 98.01 82.99 84.82 86.68 88.55 90.44 92.35 94.28 96.24 98.21 83.17 85.00 86.86 88.74 90.63 92.54 94.48 96.43 98.41 83.36 85.19 87.05 88.92 90.82 92.74 94.67 96.63 98.60 83.54 85.38 87.24 89.11 91.01 92.93 94.87 96.83 98.80 83.72 85.56 87.42 89.30 91.20 93.12 95.06 97.02 99.00 83.91 85.75 87.61 89.49 91.39 93.32 95.26 97.22 99.20 84.09 85.93 87.80 89.68 91.58 93.51 95.45 97.42 99.40 84.27 86.12 87.99 89.87 91.78 93.70 95.65 97.61 99.60 84.46 86.30 88.17 90.06 91.97 93.90 95.84 97.81 99.80 108 Powers and Roots Squares and Cubes Square Roots and Cube Roots No. Square Cube Square Eoot Cube Root No. Square Cube Square Root Cube Root 1 1 1 1.000 1.000 51 2,601 132,651 7.141 3.708 2 4 8 1.414 1.260 52 2,704 140,608 7.211 3.733 3 9 27 1.732 1.442 53 2,809 148,877 7.280 3.756 4 16 64 2.000 1.587 54 2,916 157,464 7.348 3.780 5 25 125 2.236 1.710 55 3,025 166,375 7.416 3.803 6 36 216 2.449 1.817 56 3,136 175,616 7.483 3.826 7 49 343 2.646 1.913 57 3,249 185,193 7.550 3.849 8 64 512 2.828 2.000 58 3,364 195,112 7.616 3.871 9 81 729 3.000 2.080 59 3,481 205,379 7.681 3.893 10 100 1,000 3.162 2.154 60 3,600 216,000 7.746 3.915 11 121 1,331 3.317 2.224 61 3,721 226,981 7.810 3.936 12 144 1,728 3.464 •2.289 62 3,844 238,328 7.874 3.958 13 169 2,197 3.606 2.351 63 3,969 250,047 7.937 3.979 14 196 2,744 3.742 2.410 64 4,09(5 262,144 8.000 4.000 15 225 3,375 3.873 2.466 65 4,225 274,625 8.062 4.021 16 256 4,096 4.000 2.520 66 4,356 287,496 8.124 4.041 17 289 4,913 4.123 2.571 67 4,489 300,763 8.185 4.0(32 18 324 5,832 4.243 2.621 68 4,624 314,432 8.246 4.082 19 361 6,859 4.359 2.668 69 4,761 328,509 8.307 4.102 20 400 8,000 4.472 2.714 70 4,900 343,000 8.367 4.121 21 441 9,261 4.583 2.759 71 5,041 357.911 8.426 4.141 22 484 10,648 4.690 2.802 72 5,184 373,248 8.485 4.1(50 23 529 12,167 4.796 2.844 73 5,329 389,017 8.544 4.179 24 576 13,824 4.899 2.884 74 5,476 405,224 8.602 4.198 25 625 15,625 5.000 2.924 75 5,625 421,875 8.660 4.217 26 676 17,576 5.099 2.962 76 5,776 438,976 8.718 4.236 27 729 19,683 5.196 3.000 77 5,929 456,533 8.775 4.254 28 784 21,952 5.292 3.037 78 6,084 474,552 8.832 4.273 29 841 24,389 5.385 3.072 79 6,241 493,039 8.888 4.291 30 900 27,000 5.477 3.107 80 6,400 512,000 8.944 4.309 31 961 29,791 5.568 3.141 81 6,561 531,441 9.000 4.327 32 1,024 32,768 5.657 3.175 82 6,724 551,368 9.055 4.344 33 1,089 35,937 5.745 3.208 83 6,889 571,787 9.110 4.362 34 1,156 39,304 5.831 3.240 84 7,056 592,704 9.165 4.380 35 1,225 42,875 5.916 3.271 85 7,225 614,125 9.220 4.397 36 1,296 46,656 6.000 3.302 86 7,396 636,056 9.274 4.414 37 1,369 50,653 6.083 3.332 87 7,569 658,503 9.327 4.431 38 1,444 54,872 6.164 3.362 88 7,744 681,472 9.381 4.448 39 1,521 59,319 6.245 3.391 89 7,921 704,969 9.434 4.465 40 1,600 64,000 6.325 3.420 90 8,100 729,000 9.487 4.481 41 1,681 68,921 6.403 3.448 91 8,281 753,571 9.539 4.498 42 1,764 74,088 6.481 3.476 92 8,464 778,688 9.592 4.514 43 1,849 79,507 6.557 3.503 93 8,649 804,357 9.644 4.531 44 1,936 85,184 6.633 3.530 94 8,836 830,584 9.695 4.547 45 2,025 91,125 6.708 3.557 95 9,025 857,375 9.747 4.563 46 2,116 97,336 6.782 3.583 96 9,216 884,736 9.798 4.579 47 2,209 103,823 6.856 3.609 97 9,409 912,673 9.849 4.595 48 2,304 110,592 6.928 3.634 98 9,604 941,192 9.899 4.610 49 2,401 117,649 7.000 3.659 99 9,801 970,299 9.950 4.626 50 2,500 125,000 7.071 3.684 100 10,000 1,000,000 10.000 4.642 For a more complete table, see The Macjuillan Tables, pp. 94-111. Important Constants 109 Certain Convenient Values for n = 1 to n = 10 n 1/n Vn ■y/n n\ 1/nl Logio 11 1 1.000000 1.00000 1.00000 1 1.0000000 0.000000000 2 0500000 1.41421 1.25992 2 0.5000000 0.301029996 3 0.333333 1.73205 1.44225 6 0.1666667 0.477121255 4 0.250000 2.00000 1.58740 24 0.0416667 0.602059991 5 0.200000 2.23607 1.70998 120 0.0083333 0.698970004 6 0.166667 2.44949 1.81712 720 0.0013889 0.778151250 7 0.142857 2.64575 1.91293 5040 0.0001984 0.845098040 8 0.125000 2.82843 2.00000 40320 0.0000248 0.903089987 9 0.111111 3.00000 2.08008 362880 0.0000028 0.954242509 10 0.100000 3.16228 2.15443 3628800 0.0000003 1.000000000 Logarithms of Important Constants 71 =■ NUMBER Value of n Log io n IT 3.14159265 0.49714987 1-4- 7T 0.31830989 9.50285013 7r2 9.86960440 0.99429975 VtF 1.77245385 0.24857494 e = Napierian Base 2.71828183 0.43429448 M= logw e 0.43429448 9.63778431 l-5-if=log e 10 2.30258509 0.36221569 180 -7- 7r = degrees in 1 radian 57.2957795 1.75812262 7r -r- 180 = radians in 1° 0.01745329 8.24187738 ir -4- 10800 = radians in 1' 0.0002908882 6.46372613 t -7- 648000 = radians in 1" 0.000004848136811095 4.68557487 sin 1" 0.000004848136811076 4.68557487 tan 1" 0.000004848136811152 4.68557487 centimeters in 1 ft. 30.480 1.4840158 feet in 1 cm. 0.032808 8.5159842 inches in 1 m. 39.37 (exact legal value) 1.5951654 pounds in 1 kg. 2.20462 0.3433340 kilograms in 1 lb. 0.453593 9.6566660 g (average value) 32.16 ft./sec./sec. 1.5073 = 981 cm./sec/sec 2.9916690 weight of 1 cu. ft. of water 62.425 lb. (max. density) 1.7953586 weight of 1 cu. ft. of air 0.0807 lb. (at 32° F.) 8.907 cu. in. in 1 (U. S.) gallon 231 (exact legal value) 2.3636120 ft. lb. per sec. in 1 H. P. 550. (exact legal value) 2.7403627 kg. m. per sec. in 1 H. P. 76.0404 1.8810445 watts in 1 H. P. 745.957 2.8727135 11C 1 Fo ur ] *lac e L( )gar ithr US N 1 2 3 4 5 6 7 8 9 12 3 4 5 6 7 8 9 10 0000 0043 0080 0128 0170 0212 0253 0294 0334 0374 4 8 12 17 21 25 29 33 37 11 12 13 14 15 16 17 18 19 0414 0792 1139 1461 1761 2041 2304 2553 2788 0453 0828 1173 1492 1790 2068 2330 2577 2810 0492 0864 1206 1523 1818 2095 2355 2601 2833 0531 0899 1239 1553 1847 2122 2380 2625 2856 0569 0934 1271 1584 1875 2148 2405 2648 2878 0607 0969 *1303 1614 1903 2175 2430 2672 2900 0645 1004 1335 1644 1931 2201 2455 2695 2923 0682 1038 1367 1673 1959 2227 2480 2718 2945 0719 1072 1399 1703 1987 2253 2504 2742 2967 0755 1106 1430 1732 2014 2279 2529 2765 2989 4 8 11 3 7 10 3 6 10 3 6 9 3 6 8 3 5 8 2 5 7 2 5 7 2 4 7 15 If- 23 14 17 21 13 16 19 12 15 18 11 14 17 11 13 16 10 12 15 9 12 14 9 11 13 26 30 34 24 28 31 23 26 29 21 24 27 20 22 25 18 21 24 17 20 22 16 19 21 16 18 20 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 2 4 6 8 1113 15 17 19 21 22 23 24 25 26 27 1 3222 3424 3617 3802 3979 4150 4314 4472 4624 3243 3444 3636 3820 3997 4166 4330 4487 4639 3263 3464 3655 3838 4014 4183 4346 4502 4654 3284 3483 3674 3856 4031 4200 4362 4518 4669 3304 3502 3692 3874 4048 4216 4378 4533 4683 3324 3522 3711 3892 4065 4232 4393 4548 4698 3345 3541 3729 3909 4082 .4249 4409 4564 4713 3365 3560 3747 3927 4099 4265 4425 4579 4728 3385 3579 3766 3945 4110 4281 4440 4594 4742 3404 3598 3784 3962 4133 4298 4456 4609 4757 2 4 6 2 4 6 2 4 6 2 4 5 2 4 5 2 3 5 2 3 5 2 3 5 13 4 8 10 12 8 10 12 7 9 11 7 9 11 7 9 10 7 8 10 6 8 9 6 8 9 6 7 9 14 16 18 14 16 17 13 15 17 12 14 16 12 14 16 11 13 15 11 12 14 11 12 14 10 12 13 30 31 32 33 34 35 36 37 38 39 4771 4786 4928 5065 5198 5328 5453 5575 5694 5809 5922 4800 4814 4829 4843 4857 4871 4886 4900 13 4 6 7 9 10 11 13 4914 5051 5185 5315 5441 5563 5682 5798 5911 4942 5079 5211 5340 5405 5587 5705 5821 5933 4955 5092 5224 5353 5478 5599 5717 5832 5944 4969 5105 5237 5366 5490 5611 5729 5843 5955 4983 5119 5250 5378 5502 5623 5740 5855 5966 4997 5132 5263 5391 5514 5635 5752 5866 5977 5011 5145 5276 5403 5527 5647 5763 5877 5988 5024 5159 5289 5416 5539 5658 5775 5888 5999 5038 5172 5302 5428 5551 5670 5786 5899 6010 13 4 13 4 13 4 12 4 12 4 12 4 12 4 1 2 3 1 2 3 5 7 8 5 7 8 5 7 8 5 6 8 5 6 7 5 6 7 5 6 7 5 6 7 4 5 7 10 11 12 91112 9 1112 9 10 11 9 10 11 8 1011 8 911 8 9 10 8 910 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 12 3 4 5 6 8 9 10 41 42 43 44 45 46 47 48 49 6128 6232 6335 6435 6532 6628 6721 6812 6902 6138 6213 6345 6444 6542 6637 6730 6821 6911 6149 6253 6355 6454 6551 6646 6739 6830 6920 6160 6263 6365 6464 6561 6656 6749 6839 6928 6170 6274 6375 6474 6571 6665 6758 6848 6937 6180 6284 6385 6484 6580 6675 6767 6857 6946 6191 6294 6395 6493 6590 6684 6776 6866 6955 6201 6304 6405 6503 6599 6693 6785 6875 6964 6212 6314 6415 6513 6609 6702 6794 6884 6972 6222 6325 6425 6522 6618 6712 6803 6893 6981 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 4 5 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 7 7 8 7 7 8 7 7 8 6 7 8 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 12 3 3 4 5 6 7 8 51 52 53 54 7076 7160 7243 7324 7084 7168 7251 7332 7093 7177 7259 7340 7101 7185 7267 7348 7110 7193 7275 7356 7118 7202 7284 7364 7126 7210 7292 7372 7135 7218 7300 7380 7143 7226 7308 7388 7152 7235 7316 7396 12 3 12 3 12 2 12 2 3 4 5 3 4 5 3 4 5 3 4 5 6 7 8 6 7 7 6 6 7 6 6 7 I 1 2 3 4 5 6 7 8 9 12 2 4 5 6 7 8 9 The proportional parts are stated in fall for every tenth at the right-hand side. The logarithm of any number of four significant figures can be read directly by add- Four Place Logarithms 111 If 1 2 3 4 5 6 7 8 9 12 3 4 5 6 7 8 9 55 56 57 58 59 7404 7482 7559 7634 7709 7412 7490 7566 7642 7716 7419 7497 7574 7649 7723 7427 7505 7582 7657 7731 7435 7513 7589 7664 7738 7443 7520 7597 7672 7745 7451 7528 7604 7679 7752 7459 7536 7612 7686 7760 7466 7543 7619 7694 7767 7474 7551 7627 7704 7774 12 2 12 2 1 1 2 1 1 2 112 3 4 5 34 5 3 4 5 3 4 4 3 4 4 5 6 7 5 6 7 5 6 7 5 6 7 5 6 7 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 112 3 4 4 5 6 6 61 62 63 64 65 66 67 68 69 7853 7924 7993 8062 8129 8195 8261 8325 8388 7860 7931 8000 8069 8136 8202 8267 8331 8398 7868 e 8075 8142 8209 8274 83:58 8401 7875 7945 8014 8082 8149 8215 8280 8344 8407 7882 7952 8021 8089 8156 8222 8287 8351 8414 7889 7959 8028 8096 8162 8228 8293 8357 8420 7896 7966 8035 8102 8169 8235 8299 8363 8426 7903 7910 7917 7973 7980 7987 8041 8048 8055 8109 8116 8122 8176 8182 8189 8241 8248 8254 8306 8312 8319 8370 8376 8382 8432 8439| 8445 112 1 1 2 112 112 112 112 112 112 112 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 5 6 6 5 5 6 5 5 6 5 5 6 5 5 6 5 5 6 5 5 6 4 5 6 4 5 6 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 112 3 3 4 4 5 6 71 72 73 74 75 76 77 78 79 8513 8573 8633 8692 8751 8808 8865 8921 8976 8519 8579 8639 8698 8756 8814 8871 8927 8982 8525 8585 8645 8704 8762 8820 8876 8932 8987 8531 8591 8651 8710 8768 8825 8882 8938 8993 8537 8597 8657 8716 8774 8831 8887 8943 8998 8543 8603 8663 8722 8779 8837 8893 8949 9004 8549 8609 8669 8727 8785 8842 8899 8954 9009 8555 8615 8675 87&3 8791 8848 8904 8960 9015 8561 8621 8681 8739 8797 8854 8910 8965 9020 8567 8627 8686 8745 8802 8859 8915 8971 9025 112 112 112 112 112 112 1 1 2 112 112 3 3 4 3 3 4 2 3 4 2 3 4 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 4 5 6 4 5 6 4 5 5 4 5 5 4 5 5 4 4 5 4 4 5 4 4 5 4 4 5 80 9031 9036 9042 9047 9053 9058 9063 9069| 9074 9079 1 1 2 2 3 3 4 4 5 81 82 83 84 85 86, 87 88 89 90&5 9138 9191 9243 9294 9345 9395 9445 9494 9090 9143 9196 9248 9299 9350 9400 9450 9499 9096 9149 9201 9253 9304 9355 9405 9455 9504 9101 9154 9206 9258 9309 9360 9410 9460 9509 910(5 9159 9212 9263 9315 9365 9415 9465 9513 9112 9165 9217 9269 9320 9370 9420 9469 9518 9117 9170 9222 9274 9325 9375 9425 9474 9523 9122 9175 9227 9279 9330 9380 9430 9479 9528 9128 9180 9232 9284 93,35 9385 9435 9484 9533 9133 9186 9238 9289 9340 9390 9440 9489 9538 112 112 112 112 112 112 112 Oil 1 1 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 2 3 2 2 3 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 3 4 4 3 4 4 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 958(5 Oil 2 2 3 3 4 4 91 92 93 94 95 96 97 98 99 9590 9638 9685 9731 9777 9823 9868 9912 9956 9595 9643 9689 9736 9782 9827 9872 9917 9961 9600 9647 9694 9741 9786 9832 9877 9921 9965 9605 9652 9699 9745 9791 9836 9881 9926 9969 9609 9657 9703 9750 9795 9841 9886 993(1 9974 9614 9661 9708 9754 9800 9845 9890 9934 9978 9619 9666 9713 9759 9805 9850 9894 9939 9983 9624 9671 9717 9763 9809 9854 9899 9943 9987 9628 9675 9722 9768 9814 9859 9903 9948 9991 9633 9680 9727 9773 9818 9863 9908 9952 9996 1 1 Oil Oil Oil Oil Oil 1 1 Oil 1 1 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 3 4 3 3 4 N 1 2 3 4 5 6 7 8 9 12 3 4 5 6 7 8 9 ing the proportional part corresponding to the fourth figure to the tabular number corresponding to the first three figures. There may be an error of 1 in the last place. 112 Four Place Trigonometric Functions [Characteristics of Logarithms omitted — determine by the usual rule from the value] Radians Degbees Sine Tangent Cotangent Cosine Value Log 10 Value Log 10 Value Logio Value Log 10 .0000 .0029 0°00' 10 .0000 .0029 .0000 .0029 .4637 1.0000 .0000 90° 00' -50 1.5708 1.5679 .4637 343.77 .5363 i!oooo !oooo .0058 20 .0058 .7648 .0058 .7648 171.89 .2352 1.0000 .0000 40 1.5650 .0087 30 .0087 .9408 .0087 .9409 114.59 .0591 1.0000 .0000 3Q 1.5621 .0116 40 .0116 .0658 .0116 .0658 85.940 .9342 .9999 .0000 20 1.5592 .0145 50 .0145 .1627 .0145 .1627 68.750 .8373 .9999 .0000 10 1.5563 .0175 1°00' .0175 .2419 .0175 .2419 57.290 .7581 .9998 .9999 89° 00' 1.5533 .0204 10 .0204 .3088 .0204 .3089 49.104 .6911 .9998 .9999 50 1.5504 .0233 20 .0233 .3668 .0233 .3669 42.964 .6331 .9997 .9999 40 1.5475 .0262 30 .0262 .4179 .0262 .4181 38.188 .5819 .9997 .9999 30 1.5446 .0291 40 .0291 .4637 .0291 .4638 34.368 .5362 .9996 .9998 20 1.5417 .0320 50 .0320 .5050 .0320 .5053 31.242 .4947 .9995 .9998 10 1.5388 .0349 2° 00' .0349 .5428 .0349 .5431 28.636 .4569 .9994 .9997 88° 00' 1.5359 .0378 10 .0378 .5776 .0378 .5779 26.432 .4221 .9993 .9997 50 1.5330 .0407 20 .0407 .6097 .0407 .6101 24.542 .3899 .9992 .9996 40 1.5301 .0436 30 .0436 .6397 .0437 .6401 22.904 .3599 .9990 .9996 30 1.5272 .0465 40 .0465 .6677 .0466 .6682 21.470 .3318 .9989 .9995 20 1.5243 .0495 50 .0494 .6940 .0495 .6945 20.20(3 .3055 .9988 .9995 10 1.5213 .0524 3° 00' .0523 .7188 .0524 .7194 19.081 .2806 .9986 .9994 87° 00' 1.5184 .0553 10 .0552 .7423 .0553 .7429 18.075 .2571 .9985 .9993 50 1.5155 .0582 20 .0581 .7645 .0582 .7652 17.169 .2348 .9983 .9993 40 1.5126 .0611 30 .0610 .7857 .0612 .7865 16.350 .2135 .9981 .9992 30 1.5097 .0640 40 .0640 .8059 .0641 .8067 15.605 .1933 .9980 .9991 20 1.5068 .0669 50 .0669 .8251 .0670 .8261 14.924 .1739 .9978 .9990 10 1.5039 .0698 4° 00' .0698 .8436 .0699 .8446 14.301 .1554 .9976 .9989 86° 00' 1.5010 .0727 10 .0727 .8613 .0729 .8624 13.727 .1376 .9974 .9989 50 1.4981 .0756 20 .0756 .8783 .0758 .8795 13.197 .1205 .9971 .9988 40 1.4952 .0785 30 .0785 .8946 .0787 .8960 12.706 .1040 .9969 .9987 30 1.4923 .0814 40 .0814 .9104 .0816 .9118 12.251 .0882 .9967 .9986 20 1.4893 .0844 50 .0843 .9256 .0846 .9272 11.826 .0728 .9964 .9985 10 1.4864 .0873 5° 00' .0872 .9403 .0875 .9420 11.430 .0580 .9962 .9983 85° 00' 1.4835 .0902 10 .0901 .9545 .0904 .9563 11.059 .0437 .9959 .9982 50 1.4806 .0931 20 .0929 .9682 .0934 .9701 10.712 .0299 .9957 .9981 40 1.4777 .0960 30 .0958 .9816 .0963 .9836 10.385 .0164 .9954 .9980 30 1.4748 .0989 40 .0987 .9945 .0992 .9966 10.078 .0034 .9951 .9979 20 1.4719 .1018 50 .1016 .0070 .1022 .0093 9.7882 .9907 .9948 .9977 10 1.4690 .1047 6° 00' .1045 .0192 .1051 .0216 9.5144 .9784 .9945 .9976 84° 00' 1.4661 .1076 10 .1074 .0311 .1080 .0336 9.2553 .9664 .9942 .9975 50 1.4632 .1105 20 .1103 .0426 .1110 .0453 9.0098 .9547 .9939 .9973 40 1.4603 .1134 30 .1132 .0539 .1139 .0567 8.7769 .9433 .9936 .9972 30 1.4573 .1164 40 .1161 .0648 .1169 .0678 8.5555 .9322 .9932 .9971 20 1.4544 .1193 50 .1190 .0755 .1198 .0786 8.3450 .9214 .9929 .9969 10 1.4515 .1222 7° 00' .1219 .0859 .1228 .0891 8.1443 .9109 .9925 .9968 83° 00' 1.4486 .1251 10 .1248 .0961 .1257 .0995 7.9530 .9005 .9922 .9966 /50 1.4457 .1280 20 .1276 .1060 .1287 .1096 7.7704 .8904 .9918 .9964 r40 *30 1.4428 .1309 30 .1305 .1157 .1317 .1194 7.5958 .8806 .9914 .9963 1.4399 .1338 40 .1334 .1252 .1346 .1291 7.4287 .8709 .9911 .9961 20 1.4370 .1367 50 .1363 .1345 .1376 .1385 7.2687 .8615 .9907 .9959 10 1.4341 .1396 8° 00' .1392 .1436 .1405 .1478 7.1154 .8522 .9903 .9958 82° 00' 1.4312 .1425 10 .1421 .1525 .1435 .1569 6.9682 .8431 .9899 .9956 50 1.4283 .1454 20 .1449 .1612 .1465 .1658 6.8269 .8342 .9894 .9954 40 1.4254 .1484 30 .1478 .1697 .1495 .1745 6.6912 .8255 .9890 .9952 30 1.4224 .1513 40 .1507 .1781 .1524 .1831 6.5606 .8169 .988(5 .9950 20 1.4195 .1542 50 .1536 .1863 .1554 .1915 6.4348 .8085 .9881 .9948 10 1.4166 .1571 9° 00' .1564 .1943 .1584 .1997 6.3138 .8003 .9877 .9946 81° 00' 1.4137 Value Log 10 Value Lojr 10 Value Log 10 Value Log 10 Degrees Radians Cosine Cotangent Tangent Sine Four Place Trigonometric Functions 113 [Characteristics of Logarithms omitted — ietermine by the usual rule from the value] Radians Degbees Sine Tangent Cotangent Cosine Value Log 10 Value Logio Value Log 10 Value L«g 10 .1571 9° 00' .1564 .1943 .1584 .1997 6.3138 .8003 .9877 .9946 81° 00' 1.4137 .1600 10 .1593 .2022 .1614 .2078 6.1970 .7922 .9872 .9944 50 1.4108 .1629 20 .1622 .2100 .1644 .2158 6.0844 .7842 .9868 .9942 40 1.4079 .1658 30 .1650 .2176 .1673 .2236 5.9758 .7764 L9868 '.9858 .9940 30 1.4050 .1687 40 .1679 .2251 .1703 .2313 5.8708 .7687 .9938 20 1.4021 .1716 50 .1708 .2324 .1733 .2389 5.7694 .7611 .9853 .9936 10 1.3992 .1745 10° 00 .1736 .2397 .1763 .2463 5.6713 .7537 .9848 .9934 80° 00' 1.3963 .1774 10 .1765 .2468 .1793 .2536 5.5764 .7464 .9843 .9931 50 1.3934 .1804 20 .1794 .2538 .1823 .2609 5.4845 .7391 .9838 .9929 40 1.3904 .1833 30 .1822 .2606 .1853 .2680 5.3955 .7320 .9833 .9927 30 1.3875 .1862 40 .1851 .2674 .1883 .2750 5.3093 .7250 .9827 .9924 20 1.3846 .1891 50 .1880 .2740 .1914 .2819 5.2257 .7181 .9822 .9922 10 1.3817 .1920 11°00' .1908 .2806 .1944 .2887 ,5.1446 .7113 .9816 .9919 79° 00 1.3788 .1949 10 .1937 .2870 :1974 .2953 5.0658 .7047 .9811 .9917 50 1.3759 .1978 20 .1965 .2934 .2004 .3020 4.9894 .6980 .9805 .9914 40 1.3730 .2007 30 .1994 .2997 .2035 .3085 4.9152 .6915 .9799 .9912 30 1.3701 .2036 4D .2022 .3058 .2065 .3149 4.8430 .6851 .9793 .9909 20 1.3672 .2065 50 .2051 .3119 .2095 .3212 4.7729 .6788 .9787 .9907 10 1.3643 .2094 12° 00' .2079 .3179 .2126 .3275 4.7046 .6725 .9781 .9904 78° 00' L3614 .2123 10 .2108 .3238 .2156 .3336 4.6382 .6664 .9775 .9901 50 1.3584 .2153 20 .2136 .3296 .2186 .3397 4.5736 .6603 .9769 .9899 40 1.3555 .2182 30 .2164 .3353 .2217 .3458 •4.5107 .6542 .9763 .9896 30 1.3526 1 .2211 40 .2193 .3410 .2247 .3517 4.4494 .6483 .9757 .9893 20 1.3497 .2240 50 .2221 .3466 .2278 .3576 4.3897 .6424 .9750 .9890 10 1.3468 .2269 13° 00' .2250 .3521 .2309 .3634 4.3315 .6366 .9744 .9887 77° 00' 1.3439 .2298 10 .2278 .3575 .2339 .3691 4.2747 .6309 .9737 .9884 50 1.3410 .2327 20 .2306 .3629 .2370 .3748 4.2193 .6252 .9730 .9881 40 1.3381 .2356 30 .2334 .3682 .2401 .3804 4.1653 .6196 .9724 .9878 30 1.3352 .2385 40 .2363 .3734 .2432 .3859 4.1126 .6141 .9717 .9875 20 1.3323 .2414 50 .2391 .3786 .2462 .3914 4.0611 .6086 .9710 .9872 10 1.3294 .2443 14° 00' .2419 .3837 .2493 .3968 4.0108 .6032 .9703 .9869 76° 00' 1.3265 .2473 10 .2447 .3887 .2524 .4021 3.9617 .5979 .9696 .9866 50 1.3235 .2502 20 .2476 .3937 .2555 .4074 3.9136 .5926 .9689 .9863 40 1.3206 .2531 30 .2504 .3986 .2586 .4127 3.8667 .5873 .9681 .9859 30 1.3177 .2560 40 .2532 .4035 .2617 .4178 3.8208 .5822 .9674 .9856 20 1.3148 .2589 50 .2560 .4083 .2648 .4230 3.7760 .5770 .9667 .9853 10 1.3119 .2618 15°00' .2588 .4130 .2679 .4281 3.7321 .5719 .9659 .9849 75° 00' 1.3090 .2647 10 .2616 .4177 .2711 .4331 3.6891 .5669 .9652 .9846 50 1.3061 .2676 20 .2644 .4223 .2742 .4381 3.6470 .5619 .9644 .9843 40 1.3032 .2705 30 .2672 .4269 .2773 .4430 3.6059 .5570 .9636 .9839 30 1.3003 .2734 40 .2700 .4314 .2805 .4479 3.5656 .5521 .9628 .9836 20 1.2974 .2763 50 .2728 .4359 .2836 .4527 3.5261 .5473 .9621 .9832 10 1.2945 .2793 16° 00' .2756 .4403 .2867 .4575 3.4874 .5425 .9613 .9828 74° 00' 1.2915 .2822 10 .2784 .4447 .2899 .4622 3.4495 .5378 .9605 .9825 50 1.2886 .2851 20 .2812 .4491 .2931 .4669 3.4124 .5331 .9596 .9821 40 1.2857 .2880 30 .2840 .4533 .2962 .4716 3.3759 .5284 .9588 .9817 30 1.2828 .2909 40 .2868 .4576 .2994 .4762 3.3402 .5238 .9580 .9814 20 1.2799 .2938 50 .2896 .4618 .3026 .4808 3.3052 .5192 .9572 .9810 10 1.2770 .2967 17° 00' .2924 .4659 .3057 .4853 3.2709 .5147 .9563 .9806 73° 00' 1.2741 .2996 10 .2952 .4700 .3089 .4898 3.2371 .5102 .9555 .9802 50 1.2712 .3025 20 .2979 .4741 .3121 .4943 3.2041 .5057 .9546 .9798 40 1.2683 .3054 30 .3007 .4781 .3153 .4987 3.1716 .5013 .9537 .9794 30 1.2654 .3083 40 .3035 .4821 .3185 .5031 3.1397 .4969 .9528 .9790 20 1.2625 .3113 50 .3062 .4861 .3217 .5075 3.1084 .4925 .9520 .9786 10 1.2595 .3142 18° 00' .3090 .4900 .3249 .5118 3.0777 .4882 .9511 .9782 72° 00' 1.2566 Value Logio Value Log 10 Value Log 10 Value Log 10 Degrees Radians Cosine Cotangent Tangent Sine 114 Four Place Trigonometric Functions [Characteristics of Logarith ms omitted — determine by the usual rule from the value]' Radians Degrees Sine Tangent Cotangent Cosine Value L°g 10 Value Log x Value Log 10 Value Log 10 .3142 18° 00' .3090 .4900 .3249 .5118 3.0777 .4882 .9511 .9782 72° 00' 1.2566 .3171 10 .3118 .4939 .3281 .5161 3.0475 .4839 .9502 .9778 50 1 .2537 .3200 20 .3145 .4977 .3314 .5203 3.0178 .4797 .9492 .9774 40 1.2508 .3229 30 .3173 .5015 .3346 .5245 2.9887 .4755 .9483 .9770 30 1.2479 .3258 40 .3201 .5052 .3378 .5287 2.9600 .4713 .9474 .9765 20 1.2450 .3287 50 .3228 .5090 .3411 .5329 2.9319 .4671 .9465 .9761 10 1.2421 .3316 19° 00' .3256 .5126 .3443 .5370 2.9042 .4630 .9455 .9757 71° 00' 1.2392 .3345 10 .3283 .5163 .3476 .5411 2.8770 .4589 .9446 .9752 50 1.2363 .3374 20 .3311 .5199 .3508 .5451 2.8502 .4549 .9436 .9748 40 1.2334 .3403 30 .3338 .5235 .3541 .5491 2.8239 .4509 .9426 .9743 30 1.2305 .3432 40 .3365 .5270 .3574 .5531 2.7980 .4469 .9417 .9739 20 1.2275 .3462 50 .3393 .5306 .3607 .5571 2.7725 .4429 .9407 .9734 10 1.2246 .3491 20° 00' .3420 .5341 .3640 .5611 2.7475 .4389 .9397 !9730 70° 00' 1.2217 .3520 10 .3448 .5375 .3073 .5650 2.7228 .4350 .9387 .9725 50 1.2188 .3549 20 .3475 .5409 .3706 .5689 2.6985 .4311 .9377 .9721 40 1.2159 .3578 30 .3502 .5443 .3739 .5727 2.6746 .4273 .9367 .9716 30 1.2130 .3607 40 .3529 .5477 .3772 .5766 2.6511 .4234 .9356 .9711 20 1.2101 .3636 50 .3557 .5510 .3805 .5804 2.6279 .4196 .9346 .9706 10 1.2072 .3665 21° 00' .3584 .5543 .3839 .5842 2.6051 .4158 .9336 ,9702 69° 00' 1.2043 .3694 10 .3611 .5576 .3872 .5879 2.5826 .4121 .9325 .9697 50 1.2014 .3723 20 .3638 .5609 .3906 .5917 2.5605 .4083 .9315 .9692 40 1.1985 .3752 30 .3665 .5641 .3939 .5954 2.5386 .4046 .9304 .9687 30 1.1956 .3782 40 .3692 .5673 .3973 .5991 2.5172 .4009 .9293 .9682 20 1.1926 .3811 50 .3719 .5704 .4006 .6028 2.4960 .3972 .9283 ,.9677 10 1.1897 .3840 22° 00' .3746 .5736 .4040 .6064 2.4751 .3936 .9272 .9672 68° 00' 1.1868 .3869 10 .3773 .5767 .4074 .6100 12.4545 12.4342 .3900 .9261 .9667 50 1.1839 .3898 20 .3800 .5798 .4108 .6136 .3864 .9250 .9661 40 1.1810 .3927 30 .3827 .5828 .4142 .6172 2.4142 .3828 .9239 .9656 30 1.1781 .3956 40 .3854 .5859 .4176 .6208 2.3945 .3792 .9228 .9651 20 1.1752 .3985 50 .3881 .5889 .4210 .6243 2.3750 .3757 .9216 .9646 10 1.1723 .4014 23° 00' .,3907 .5919 .4245 .6279 2.3559 .3721 .9205 .9640 67° 00' 1.1694 .4043 10 .3934 .5948 .4279 .6314 2.3369 .3686 .9194 .9635 50 1.1665 .4072 20 .3961 .5978 .4314 .6348 2,3183 .3652 .9182 .9629 40 1.163(5 .4102 30 .3987 .6007 .4348 .6383 2.2998 .3617 .9171 .9624 30 1.1606 .4131 40 .4014 .6036 .4383 .6417 2.2817 .3583 .9159 .9618 20 1.1577 .4160 50 .4041 .6065 .4417 .6452 2.2637 .3548 .9147 .9613 1Q 1.1548 .4189 24° 00' .4067 .6093 .4452 .6486 2.24(50 .3514 .9135 .9607 66° 00' 1.1519 .4218 10 .4094 .6121 .4487 .6520 2.2286 .3480 .9124 .9602 50 1.1490 .4247 20 .4120 .6149 .4522 .6553 2.2113 .3447 .9112 .9596 40 1.1461 .4276 30 .4147 .6177 .4557 .6587 2.1943 .3413 .9100 .9590 30 1.1432 .4305 40 .4173 .6205 .4592 .6620 2.1775 .3380 .9088 .9584 20 1.1403 .4334 50 .4200 .6232 .4628 .6654 2.1609 .3346 .9075 .9579 10 1.1374 .4363 25° 00' .4226 .6259 .4663 .6687 2.1445 .3313 .9063 .9573 65° 00' 1.1345 .4392 10 .4253 .6286 .4699 .6720 2.1283 .32&0 .9051 .9567 50 1.1316 .4422 20 .4279 .6313 .4734 .6752 2.1123 .3248 .9038 .9561 40 1.1286 .4451 30 .4305 .6340 .4770 .6785 2.0965 .3215 .9026 .9555 30 1.1257 .4480 40 .4331 .6366 .4806 .6817 2.0809 .3183 .9013 .9549 20 1.1228 .4509 50 .4358 .6392 .4841 .6850 2.0655 .3150 .9001 .9543 10 1.1199 .4538 26° 00' .4384 .6418 .4877 .6882 2.0503 .3118 .8988 .9537 64° 00' 1.1170 .4567 10 .4410 .6444 .4913 .6914 2.0353 .3086 .8975 .9530 50 1.1141 .4596 20 .4436 .6470 .4950 .6946 2.0204 .3054 .8962 .9524 40 1.1112 .4625 30 .4462 .6495 .4986 .6977 2.0057 .3023 .8949 .9518 30 1.1083 .4654 40 .4488 .6521 .5022 .7009 1.9912 .2991 .893(5 .9512 20 1.1054 .4683 50 .4514 .6546 .5059 .7040 1.9768 .2960 .8923 .9505 10 1.1025 .4712 27° 00' .4540 .6570 .5095 .7072 1.9626 .2928 .8910 .9499 63° 00' 1.0996 Value Logio Value LOftfl Value Loffio Value ' Log 10 Degrees Radians Cosine Cotangent . Tanoent . Sine Four Place Trigonometric Functions 115 [Characteristi cs of Logarithms omitted — determine by the usual rule from the value] Radians Degeees SlXE Value Log 10 Tangent Cotangent Cosine Value Log 10 Value Log 10 i Value Log 10 .4712 27° 00' .4540 .6570 .5095 .7072 1.9626 .2928 .8910 .9499 63° 00' 1.0996 .4741 10 .4566 .6595 .5132 .7103 1.94S6 .2897 .8897 .9492 50 1.0966 .4771 20 .4592 .6620 .5169 .7134 1.9347 .2866 .8884 .9486 40 1.0937 .4800 30 .4617 .6644 .5206 .7165 1.9210 .2835 .8870 .9479 30 1.0908 .4829 40 .4643 .6668 .5243 .7196 1.9074 .2804 .8857 .9473 20 1.0879 .4858 50 .4669 .6692 .5280 .7226 1.8940 .2774 .8843 .9466 10 1.0850 .4887 28° 00' .4695 .6716 .5317 .7257 1.8807 .2743 .8829 .9459 62° 00' 1.0821 .4916 10 .4720 .6740 .5354 .7287 1.8676 .2713 .8816 .9453 50 1.0792 .4945 20 .4746 .6763 .5392 .7317 1.8546 .2683 .8802 .9446 40 1.0703 .4974 30 .4772 .6787 .5430 .7348 1.8418 .2652 .8788 .9439 30 1.0734 ,5003 40 .4797 .6810 .5167 .7378 1.8291 .2622 .8774 .9432 20 1.0705 .5032 50 .4823 .6833 .5505 .7408 1.8165 .2592 .8760 .9425 10 1.0676 .5061 29° 00' .4848 .6856 .5543 .7438 1.8040 .2562 .8746 .9418 61° 00' 1.0647 .5091 10 .4874 .6878 .5581 .7467 1.7917 .2533 .8732 .9411 50 1.0617 .5120 20 .4899 .6901 .5619 .7497 1.7796 .2503 .8718 .9404 40 1.0688 .5149 30 .4924 .6923 .5658 .7526 1.7675 .2474 .8704 .9397 30 1.0559 .5178 40 .4950 .6946 .5696 .7556 1.7556 .2444 .8689 .9390 20 1.0530 .5207 50 .4975 .6968 .5735 .7585 1.7437 .2415 .8675 .9383 10 1.0501 .5230 30° 00' .5000 .6990 .5774 .7614 1.7321 .2386 .8660 .9375 60° 00' 1.0472 .5265 10 .5025 .7012 .5812 .7044 1.7205 .2356 .8646 .9368 50 1.0443 .5294 20 .5050 .7033 ^5851 .7673 1.7090 .2327 .8631 .9361 40 1.0414 .5323 30 .5075 .7055 .3890 .7701 1.6977 .2299 .8616 .9353 30 1.0385 .5352 40 .5100 .7076 .5930 .7730 1.6864 .2270 .8601 .9346 20 1.0356 .5381 50 .5125 .7097 .5969 .7759 1.6753 .2241 .8587 .9338 10- 1.0327 .5411 31° 00' .5150 .7118 .6009 .7788 1.6643 .2212 .8572 .9331 59° 00' 1.0297 .5440 10 .5175 .7139 .6048 .7816 1.6534 .2184 .8557 .9323 50 1.0268 .5469 20 .5200 .7160 .6088 .7845 1.6426 .2155 .8542 .9315 40 1.0239 .5498 30 .5225 .7181. .6128 .7873 1.6319 .2127 .8526 .9308 _30 1.0210 .5527 40 .5250 .7201 .6168 .7902 1.6212 .2098 .8511 .9300 20 1.0181 .5556 10 .5275 .7222 .6208 .7930 1.6107 .2070 .8496 .9292 10 1.0152 .5585 32° 00' .5299 .7242 .6249 .7958 1.6003 .2042 .8480 .9284 58° 00' 1.0123 .5(314 10 .5324 .7262 .6289 .7986 1.5900 .2014 .8465 .9276 50 1.0094 .5643 20 .5318 .7282 .6330 .8014 1.5798 .1986 .8450 .9268 40 1.0065 .5672 30 .5373 .7302 .6371 .8042 1.5697 .1958 .8434 .9260 30 1.0036 .5701 40 .5398 .7322 .6412 .8070 1.5597 .1930 .8418 .9252 20 1.0007 .5730 50 .5422 .7342 .6453 .8097 1.5497 .1903 .8403 .9244 10 .9977 .5760 '33° 00' .5446 .7361 .6494 .8125 1.5399 .1875 .8387 .9236 57° 00' .9948 .5789 10 .5471 .7380 .6536 .8153 1.5301 .1847 .8371 .9228 50 .9919 .5818 20 .5495 .7400 .6577 .8180 1.5204 .1820 1^8355 .9219 «8339 .9211 40 .9890 .5847 30 .5519 .7419 .6619 .8208 1.5108 .1792 30 .9861 .5876 40 .5544 .7438 .6661 .8235 1.5013 .1765 .8323 .9203 20 .9832 .5905 50 .5568 .7457 .6703 .8263 1.4919 .1737 .8307 .9194 10 .9803 .5934 34° 00' .5592 .7476 .6745 .8290 1.4826 .1710 .8290 .9186 56° 00' .9774 .5963 10 .5616 .7494 .6787 .8317 1.4733 .1683 .8274 .9177 50 .9745 .5992 20 .5640 .7513 .6830 .8344 1.4641 .1656 .8258' .9169 40 .9716 .6021 30 .5664 .7531 .6873 .8371 1.4550 .1629 .8241 .9160 30 .9687 .6050 40 .5688 .7550 .6916 .8398 1.4460 .1602 .8225 .9151 20 .9657 .6080 50 .5712 .7568 .6959 .8425 1.4370 .1575 .8208 .9142 10 .9628 .6109 35° 00' .5736 .7586 .7002 .8452 1.4281 .1548 .8192 .9134 55° 00' .9599 .6138 10 .5760 .7604 .7046 .8479 1.4193 .1521 .8175 .9125 50 .9570 .6167 20 .5783 .7622 .7089 .8506 1.4106 .1494 .8158 .9116 40 .9541 .6196 30 .5807 .7640 .7133 .8533 1.4019 .1467 .8141 .9107 30 .9512 .6225 40 .5831 .7657 .7177 .8559 1.3934 .1441 .8124 .9098 20 .9483 .6254 50 .5854 .7675 .7221 .8586 1.3848 .1414 .8107 .9089 10 .9454 .6283 36° 00' .5878 .7692 .7265 .8613 1.3764 .1387 .8090 .9080 54° 00' .9425 Value Log 10 Value Loer 10 Value Log 10 Value Log 10 Degrees Radians Cosine Cotangent Tangent Sine 116 Four Place Trigonometric Functions [Characteristics of Logarithms omitted — determine by the usual rule from the value] Radians Degress Sine Tangent Cotangent Cosine Value Log 10 Value Log 1( Value Log 10 Value Log lf .6283 36° 00' .5878 .7602 .7265 .8613 1.3704 .1387 .8090 .9080 54° 00' .9425 .6312 10 .5901 .7710 .7310 .8639 1.3080 .1301 .8073 .9070 50 .9390 .6341 20 .5925 .7727 .7355 .8666 1.3597 .1334 ,8050 .9001 40 .9307 .6370 30 .5948 .7744 .7400 .8692 1.3514* .1308 1.3432 7 '.1282 [8039 .9052 30 .9338 .6400 40 .5972 .7761 .7445 .8718 ..8021 .9042 20 .9308 .6429 50 .5995 .7778 .7490 .8745 1.3351 .1255 .8004 .9033 10 .9279 .6458 37° 00' .6018 .7795 .7536 .8771 1.3270 .1229 .7980 .9023 53° 00' .9250 .6487 10 .6041 .7811 .7581 .8797 1.3190 .1203 .7909 .9014 50 .9221 .6516 20 .6065 .7828 .7627 .8824 1.3111 .1170 17951 .9004 [7934 .8995 40 .9192 .6545 30 .6088 .7844 .7673 .8850 1.3032 .1150 30 .9103 .6574 40 .6111 .7861 .7720 .8870 1.2954 .1124 .7910 .8985 20 .9134 .6603 50 .6134 .7877 .7766 .8902 1.2876 .1098 .7898 .8975 10 .9105 .6632 38° 00' .6157 .7893 .7813 .8928 1.2799 .1072 .7880 .8905 52° 00' .9070 .6661 10 .6180 .7910 .7860 .8954 1.27231 .1046 1:2647/ .1020 .7802 .8955 50 .9047 .6690 20 .6202 .7926 ,.7907 .8980 '.7954 .9006 .7844 .8945 ~40 .9018 .6720 30 .6225 .7941 1.2572 .0994 .7820 .8935 -^30 .8988 .6749 .6778 40 50 .6248 .7957 .6271 .7973 ,£002 .9032 .8050 .9058 1.24971 .0908 1.24221.0942 .7808 .8925 .7790 .8915 ^■20 10 .8959 .8930 .6807 39° 00' .6293 .7989 .8098 .9084 1.2349 .0910 .7771 .8905 51°00' .8901 .6836 10 .6316 .8004 .8146 .9110 1.2270 .0890 .7753 .8895 50 .8872 .6865 20 .6338 .8020 .8195 .9135 1.2203 .0805 .7735 .8884 40 .8843 .6894 30 .6361 .8035 .8243 .9161 1.2131 .0839 .7710 .8874 30 .8814 .6923 40 .6383 .8050 .8292 .9187 1.2059 .0813 .7098 .8804 20 .8785 .6952 50 .6406 .8066 .8342 .9212 1.1988 .0788 .7079 .8853 10 .8750 .6981 40° 00' .6428 .8081 .8391 .9238 1.1918 .0762 .7000 .8843 50° 00' .8727 .7010 10 .6450 .8096 .8441 .9264 1.1847 .0736 .7042 .8832 50 .8098 .7039 20 .6472 .8111 .8491 .9289 1.1778 .0711 .7023 .8821 40 .8008 .7069 30 .6494 .8125 .8541 .9315 1.1708 .0685 .7604 .8810 30 .8039 .7098 40 .6517 .8140 .8591 .9341 1.1640 .0659 .7585 .8800 20 .8010 .7127 50 .6539 .8155 .8642 .9366 1.1571 .0634 .7566 .8789 10 .8581 .7156 41° 00' .6561 .8169 .8693 .9392 1.1504 .0608 .7547 .8778 49° 00' .8552 .7185 10 .6583 .8184 .8744 .9417 1.1436. .0583 .7528 .8767 50 .8523 .7214 20 .6604 .8198 .8796 .9443 1.1369 .0557 .7509 .8756 40 .8494 .7243 30 .6626 .8213 .8847 .9468 1.1303 .0532 1. 1237 .0506 .7490 .8745 30 .8405 .7272 40 .6648 .8227 .8899 .9494 .7470 .8733 20 .8430 .7301 50 .6670 .8241 .8952 .9519 1.1171 .0481 .7451 .8722 10 .8407 .7330 42° 00' .6691 .8255 .9004 .9544 1.1100 .0450 .7431 .8711 48° 00' .8378 .7359 10 .6713 .8269 .9057 .9570 1.1041 .0430 .7412 .8699 50 .8348 .7389 20 .6734 .8283 .9110 .9595 1.0977 .0405 .7392 .8688 40 .8319 .7418 30 .6756 .8297 —9163 .9021 .9217 .9(340 1.0913 .0379 .7373 .8676 ,30 .8290 .7447 40 .6777 .8311 1.0850. 0354 .7353 .8605 20 .8201 .7476 50 .6799 .8324 .9271 .9071 1.0780 .0329 ^7333 .8653 .7314 .8641 10 .8232 .7505 43° 00' .6820 .8338 .9325 .9097 1.0724 .0303 47° 00' .8203 .7534 10 .6841 .8351 .9380 .9722 1.0001 .0278 .7294 .8629 50 .8174 .7563 20 .6862 .8365 .9435 .9747 1.0599 .0253 .7274 .8018 40 .8145 .7592 30 .688*^8378 .9490 .9772 1.0538 .0228 1.0477 .0202 .7254 .8000 30 .8110 .7621 40 .6905 .8391 .9545 .9798 .7234 .8594 20 .8087 .7650 50 .6926 .8405 .9001 .9823 1.0410 .0177 .7214 .8582 10 .8058 .7679 44° 00' .6947 .8418 .9057 .9848 1.0355 .0152 .7193 .8509 46° 00' .8029 .7709 10 .6967 .8431 .9713 .9874 1.0295 .0120 .7173 .8557 50 .7999 .7738 20 ,6988 .8444 .9770 .9899 1.0235 .0101 .7153 .8545 40 .7970 .7767 30 .7009 .8457 .9827 .9924 1.0170 .0070 .7133 .8532 30 .7941 .7796 40 .7030 .8469 .9884 .9949 1.0117 .0051 .7112 .8520 20 .7912 .7825 50 .7050 .8482 .9942 .9975 1.0058 .0025 .7092 .8507 10 .7883 .7854 45° 00' .7071 .8495 1.0000 .0000 1.0000 .0000 .7071 .8495 45° 00' .7854 Value Log 10 Value Log 10 Value Log 10 Value Log 10 Degrees EIadians Cosine { Cotangent Tangent Sine Tallies and Logarithms of Haversines 117 [Characteristics of Logarithms omitted - — determine by rule from the value] 10' 20' 3C ' 40' . 5C ' Value Log 10 Value Log 10 Value Log 10 Value Log 10 Value Log 10 Value Log 10 .0000 .0000 4.3254 .0000 4.9275 .0000 5.2796 .0000 5.5295 .0001 5.7233 .0001 5.8817 .0001 6.0156 .0001 6.1315 .0002 .2338 .0002 .3254 .0003 .4081 2 .0003 .4837 .0004 .5532 .0004 .6176 .0005 .6775 .0005 .7336 .0006 .7862 3 .0007 .8358 .0008 .8828 .0008 .9273 .0009 .9697 .0010 .0101 .0011 .0487 4 .0012 .0856 .0013 .1211 .0014 .1551 .0015 .1879 .0017 .2195 .0018 .2499 5 .0019 .2793 .0020 .3078 .0022 .3354 .0023 .3621 .0024 .3880 .0026 .4132 6 .0027 .4376 .0029 .4614 .0031 .4845 .0032 .5071 .0034 .5290 .0036 .5504 7 .0037 .5713 .0039 .5918 .0041 .6117 .0043 .6312 .0045 .6503 .0047 .6689 8 .0049 .6872 .0051 .7051 .0053 .7226 .0055 .7397 .0057 .7566 .0059 .7731 9 .0062 .7893 .0064 .8052 .0066 .8208 .0069 .8361 .0071 .8512 .0073 .8660 10 .0076 .8806 .0079 .8949 .0081 .9090 .0084 .9229 .0086 .9365 .0089 .9499 11 .0092 .9631 .0095 .9762 .0097 .9890 .0100 .0016 .0103 .0141 .0106 .0264 12 .0109 .0385 .0112 .0504 .0115 .0622 .0119 .0738 .0122 .0853 .0125 .0966 13 .0128 .1077 .0131 .1187 .0135 .1296 .0138 .1404 .0142 .1510 .0145 .1614 14 .0149 .1718 .0152 .1820 .0156 .1921 .0159 .2021 .0163 .2120 .0167 .2218 15 .0170 .2314 .0174 .2409 .0178 .2504 .0182 .2597 .0186 .2689 .0190 .2781 16 .0194 .2871 .0198 .2961 .0202 .3049 .0206 .3137 .0210 .3223 .0214 .3309 17 .0218 .3394 .0223 .3478 .0227 .3561 .0231 .3644 .0236 .3726 .0240 .3806 18 .0245 .3887 .0249 .3966 .0254 .4045 .0258 .4123 .0263 .4200 .0268 .4276 19 .0272 .4352 .0277 .4427 .0282 .4502 .0287 .4576 .0292 .4649 .0297 .4721 20 .0302 .4793 .0307 .4865 .0312 .4936 .0317 .5006 .0322 .5075 .0327 .5144 21 .0332 .5213 .0337 .5281 .0343 .5348 .0348 .5415 .0353 .5481 .0359 .5547 22 .0364 .5612 .0370 .5677 .0375 .5741 .0381 .5805 .0386 .5868 .0392 .5931 23 .0397 .5993 .0403 .6055 .0409 .6116 .0415 .6177 .0421 .6238 .0426 .6298 24 .0432 .6357 .0438 .6417 .0444 .6476 .0450 .6534 .0456 .6592 .0462 .6650 25 .0468 .6707 .0475 .6764 .0481 .6820 .0487 .6876 .0493 .6932 .0500 .6987 26 .0506 .7042 .0512 .7096 .0519 .7151 .0525 .7204 .0532 .7258 .0538 .7311 27 .0545 .7364 .0552 .7416 .0558 .7468 .0565 .7520 .0572 .7572 .0578 .7623 28 .0585 .7673 .0592 .7724 .0599 .7774 .0606 .7824 .0613 .7874 .0620 .7923 29 .0627 .7972 .0634 .8020 .0641 .8069 .0648 .8117 .0655 .8165 .0663 .8213 30 .0670 .8260 .0677 .8307 .0684 .8354 .0692 .8400 .0699 .8446 .0707 .8492 31 .0714 .8538 .0722 .8583 .0729 .8629 .0737 .8673 .0744 .8718 .0752 .8763 32 .0760 .8807 .0767 .8851 .0775 .8894 .0783 .8938 .0791 .8981 .0799 .9024 33 .0807 .9067 .0815 .9109 .0823 .9152 .0831 .9194 .0839 .9236 .0847 .9277 34 .0855 .9319 .0863 .9360 .0871 .9401 .0879 .9442 .0888 .9482 .0896 .9523 35 .0904 .9563 .0913 .9603 .0921 .9643 .0929 .9682 .0938 .9722 .0946 .9761 36 .0955 .9800 .0963 .9838 .0972 .9877 .0981 .9915 .0989 .9954 .0998 .9992 37 .1007 .0030 .1016 .0067 .1024 .0105 .1033 .0142 .1042 .0179 .1051 .0216 38 .1060 .0253 .1069 .0289 .1078 .0326 .1087 .0362 .1096 .0398 .1105 .0434 39 .1114 .0470 .1123 .0505 .1133 .0541 .1142 .0576 .1151 .0611 .1160 .0646 40 .1170 .0681 .1179 .0716 .1189 .0750 .1198 .0784 .1207 .0817 .1217 .0853 41 .1226 .0887 .1236 .0920 .1246 .0954 .1255 .0987 .1265 .1021 .1275 .1054 42 .1284 .1087 .1294 .1119 .1304 .1152 .1314 .1185 .1323 w .1217 .1333 .1249 43 .1343 .1282 .1353 .1314 .1363 .1345 .1373 .1377 .1383'. 1409 .1393 .1440 44 .1403 .1472 .1413 .1503 .1424 .1534 .1434 .1565 .1444 .1596 .1454 .1626 45 .1464 .1657 .1475 .1687 .1485 .1718 .1495 .1748 .1506 .1778 .1516 .1808 46 .1527 .1838 .1538 .1867 .1548 .1897 .1558 .1926 .1569 .1956 .1579 .1985 47 .1590 .2014 .1600 .2043 .1611 .2072 .1622 .2101 .1633 .2129 .1644 .2158 48 .1654 .2186 .1665 .2215 .1676 .2243 .1687 .2271 .1698 .2299 .1709 .2327 49 .1720 .2355 1731 .2382 .1742 .2410 .1753 .2437 .1764 .2465 .1775 .2492 50 .1786 .2519 .1797 .2546 .1808 .2573 .1820 .2600 .1831 .2627 .1842 .2653 51 .1853 .2680 .1865 .2700 .1876 .2732 .1887 .2759 .1899 .2785 .1910 .2811 52 .1922 .2837 .1933 .2863 .1945 .2888 .1956 .2914 .1968 .2940 .1979 .2965 53 .1991 .2991 .2003 .3016 .2014 .3041 .2026 .3066 .2038 :3091 .2049 .3116 54 .2061 .3141 .2073 .3166 .2085 .3190 .2096 .3215 .2108 .3239 .2120 .3264 55 .2132 .3288 .2144 .3312 .2156 .3336 .2168 .3361 .2180 .3384 .2192 .3408 56 .2204 .3432 .2216 .3456 .2228 .3480 .2240 .3503 .2252 .3527 .2265 .3550 57 .2277 .3573 .2289 .3596 .2301 .3620 .2314 .3643 .2326 .3666 .2338 .3689 58 .2350 .3711 .2363 .3734 .2375 .3757 .2388 .3779 .2400 .3802 .2412 .3824 59 .2425 .3847 | .2437 .3869 .2450 .3891 .2462 .3913 .2475 .3935 .2487 .3957 118 Values and Logarithms of Haversines [Characteristics of Logarithms omitted — determine by rule from the value] 60 61 62 63 64 65 66 67 68 69 70 71 72 78 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 0' Value Log 10 .2500 .2576 .2653 .2730 .2808 .2887 .2966 .3046 .3127 .3208 .3290 .3372 .3455 .3538 .3622 .3706 .3790 .3875 .3960 .4046 .4132 .4218 .4304 .4391 .4477 .4564 .4651 .4738 .4826 .4913 .5000 .5087 .5174 .5262 .5349 .5436 .5523 .5609 .5696 .5782 .5868 .5954 .6040 .6125 .6210 .6294 .6378 .6462 .6545 .6628 .6710 .6792 .6873 .6954 .7034 .7113 .7192 .7270 .7347 .7424 .3979 .4109 .4237 .4362 .4484 .4604 .4722 .4838 .4951 .5063 .5172 .5279 .5384 .5488 .5589 .5689 .5787 .5883 .5977 .6070 .6161 .6251 .6339 .6425 .6510 .6594 .6676 .6756 .6835 .6913 .6990 .7065 .7139 .7211 .7283 .7353 .7421 .7489 .7556 .7621 .7685 .7748 .7810 .7871 .7931 .7989 .8047 .8104 .8159 .8214 .8267 .8320 .8371 .8422 .8472 .8521 .8568 .8615 .8661 .8706 10' Value Log 10 .2513 .2589 .2665 .2743 .2821 .2900 .2980 .3060 .3140 .3222 .3304 .3386 .3469 .3552 .3636 .3720 .3805 .3889 .3975 .4060 .4146 .4232 .4319 .4405 .4492 .4579 .4753 .4840 .4937 .5015 .5102 .5189 .5276 .5363 .5450 .5537 .5624 .5710 .5797 .5883 .5968 .6054 .6139 .6224 .6308 .6392 .6476 .6559 .6642 .6724 .6805 .6887 .6967 .7047 .7126 .7205 .7283 .7360 .7437 .4001 .4131 .4258 .4382 .4504 .4624 .4742 .4857 .4970 .5081 .5190 .5297 .5402 .5505 .5606 .5705 .5803 .5899 .5993 .6085 .6176 .6266 .6353 .6440 .6524 .6607 .6689 .6770 .6848 .6926 .7002 .7077 .7151 .7223 .7294 .7364 .7433 .7500 .7567 .7632 .7696 .7759 .7820 .7881 .7940 .7999 .8056 .8113 .8168 .8223 .8276 .8329 .8380 .8430 .8480 .8529 .8576 .8623 .8669 .8714 20' Value Log t 30' Value Log 10 .2525 .4023 .2601 .4152 .2678 .4279 .2756 .4403 .2834 .4524 .2913 .4644 .2993 .4761 .3073 .4876 .3154 .4989 .3235 .5099 .3317 .5208 .3400 .5314 .3483 .5419 .3566 .5522 .3650 .5623 .3734 .5722 .3819 .5819 .3904 .5915 .3989 .6009 .4075 .6101 .4160 .6191 .4247 .6280 .4333 .6368 .4420 .6454 .4506 .6538 .4593 .6621 .4680 .6703 .4767 .6783 .4855 .6862 .4942 .6939 .5029 .7015 .5116 .7090 .5204 .7163 .5291 .7235 .5378 .7306 .5465 .5552 .5638 .5725 .5811 .5897 .5983 .6068 .6153 .6238 .6322 .6406 .6490 .6573 .6655 .6737 .6819 .6900 ,6980 .7060 .7139 .7218 .7296 .7373 .7449 .7376 .7444 .7511 .7577 .7642 .7706 .7769 .7830 .7891 .7950 .8009 .8122 .8177 .8232 .8285 .8337 .8388 .8439 .8488 .8537 .8584 .8631 .8676 .8721 .2538 .2614 .2691 .2769 .2847 .2927 .3006 .3087 .3167 .3249 .3331 .3413 .3496 .3580 .3664 .3748 .3833 .3918 .4003 .4089 .4175 .4261 .4347 .4434 .4521 .4608 .4695 .4782 .4869 .4956 .5044 .5131 .5218 .5305 .5392 .5479 .5566 .5653 .5739 .5825 .5911 .5997 .6082 .6167 .6252 .6336 .6420 .6504 .6587 .6669 .6751 .6833 .6913 .6994 .7073 .7153 .7231 .7309 .7386 .7462 .4045 .4173 .4300 .4423 .4545 .4664 .4780 .4895 .5007 .5117 .5226 .5332 .5436 .5539 .5639 .5738 .5835 .5930 .6024 .6116 .6206 .6295 .6382 .6468 .6552 .6635 .6716 .6796 .6875 .6952 .7027 .7102 .7175 .7247 .7318 .7387 .7455 .7523 .7588 .7653 .7717 .7779 .7841 .7901 .7960 .8018 .8075 .8131 .8187 .8241 .8294 .8346 .8397 .8447 .8496 .8545 .8592 .8638 .8684 .8729 40' Value Log 10 .2551 .2627 .2704 .2782 .2861 .2940 .3020 .3100 .3181 .3263 .3345 .3427 .3510 .3594 .3678 .3762 .3847 .3932 .4017 .4103 .4189 .4275 .4362 .4448 .4535 .4622 .4709 .4796 .4884 .4971 .5058 .5145 .5233 .5320 .5407 .5494 .5580 .5667 .5753 .5840 .5925 .6011 .6096 .6181 .6266 .6350 .6434 .6517 .6600 .6683 .6765 .6846 .6927 .7007 .7087 .7166 .7244 .7322 .7399 .7475 .4066 .4195 .4320 .4444 .4565 .4683 .4799 .4914 .5026 .5136 .524*4 .5349 .5454 .5556 .5656 .5754 .5851 .5946 .6039 .6131 .6221 .6310 .6397 .6482 .6566 .6649 .6730 .6809 .6887 .6964 .7040 .7114 .7187 .7259 .7329 .7399 .7467 .7534 .7599 .7664 .7727 .7790 .7851 .7911 .7970 .8028 .8085 .8141 .8196 .8250 .8302 .8354 .8405 .8455 .8504 .8553 .8600 .8646 .8691 .8736 50' Value Log 10 .2563 .2640 .2717 .2795 .2874 .2953 .3033 .3113 .3195 .3276 .3358 .3441 .3524 .3608 .3692 .3776 .3861 .3946 .4032 .4117 .4203 .4290 .4376 .4463 .4550 .4637 .4724 .4811 .4898 .4985 .5073 .5160 .5247 .5334 .5421 .5508 .5595 .5682 .5768 .5854 .5940 .6025 .6111 .6195 .6280 .6364 .6448 .6531 .6614 .6696 .6778 .6860 .6940 .7020 .7100 .7179 .7257 .7335 .7411 .7487 .4088 .4216 .4341 .4464 .4584 .4703 .4819 .4932 .5044 .5154 .5261 .5367 .5471 .5572 .5672 .5771 .5867 .5962 .6055 .6146 .6236 .6324 .6411 .6496 .6580 .6662 .6743 .6822 .6900 .6977 .7052 .7126 .7199 .7271 .7341 .7410 .7478 .7545 .7610 .7674 .7738 .7800 .7861 .7921 .7980 .8037 .8094 .8150 .8205 .8258 .8311 .8363 .8414 .8464 .8513 .8561 .8608 .8654 .8699 .8743 Values and Logarithms of Haversines [Characteristics of Logarithms omitted — determine by rule from the value] 119 . C 10' 20' 30' 40' 5)' Value Log w Value Logi Value Log 10 Value Log 10 Value Log 10 Value Log 10 120 .7500 .8751 .7513 .8758 .7525 .8765 .7538 .8772 .7550 .8780 .7563 .8787 121 .7575 .8794 .7588 .8801 .7600 .8808 .7612 .8815 .7625 .8822 .7637 .8829 122 .7650 .8836 .7662 .8843 .7674 .8850 .7686 .8857 .7699 .8864 .7711 .8871 123 .7723 .8878 .7735 .8885 .7748 .8892 .7760 .8898 .7772 .8905 .7784 .8912 124 .7796 .8919 .7808 .8925 .7820 .8932 .7832 .8939 .7844 .8945 .7856 .8952 125 .7868 .8959 .7880 .8965 .7892 .8972 .7904 .8978 .7915 .8985 .7927 .8991 126 .7939 .8998 .7951 .9004 .7962 .9010 .7974 .9017 .7986 .9023 .7997 .9030 127 .8009 .9036 .8021 .9042 .8032 .9048 .8044 .9055 .8055 .9061 .8067 .9067 128 .8078 .9073 .8090 .9079 .8101 .9085 .8113 .9092 .8124 .9098 .8135 .9104 129 .8147 .9110 .8158 .9116 .8169 .9122 .8180 .9128 .8192 .9134 .8203 .9140 130 .8214 .9146 .8225 .9151 .8236 .9157 .8247 .9163 .8258 .9169 .8269 .9175 131 .8280 .9180 .8291 .9186 .8302 .9192 .8313 .9198 .8324 .9203 .8335 .9209 132 .8346 .9215 .8356 .9220 .8367 .9226 .8378 .9231 .8389 .9237 .8399 .9242 133 .8410 .9248 .8421 .9253 .8431 .9259 .8442 .9264 .8452 .9270 .8463 .9275 134 .8473 .9281 .8484 .9286 .8494 .9291 .8501 .9297 .8515 .9302 .8525 .93p7 135 .8536 .9312 .8546 .9318 .8556 .9323 .8566 .9328 .8576 .9333 .8587 .9338 136 .8597 .9343 .8607 .9348 .8617 .9353 .8627 .9359 .8637 .9364 .8647 .9369 137 .8657 .9374 .8667 .9379 .8677 .9383 .8686 .9388 .8696 .9393 .8706 .9398 138 .8716 .9403 .8725 .9408 .8735 .9413 .8745 .9417 .8754 .9422 .8764 .9427 139 .8774 .9432 .8783 .9436 .8793 .9441 .8802 .9446 .8811 .9450 .8821 .9455 140 .8830 .9460 .8840 .9464 .8849 .9469 .8858 .9473 .8867 .9478 .8877 .9482 141 .8886 .9487 .8895 .9491 .8904 .9496 .8913 .9500 .8922 .9505 .8931 .9509 142 .8940 .9513 .8949 .9518 .8958 .9522 .8967 .9526 .8976 .9531 .8984 .9535 143 .8993 .9539 .9002 .9543 .9011 .9548 .9019 .9552 .9028 .9556 .9037 .9560 144 .9045 .9564 .9054 .9568 .9062 .9572 .9071 .9576 .9079 .9580 .9087 .9584 145 .9096 .9588 .9104 .9592 .9112 .9596 .9121 .9600 .9129 .9604 .9137 .9608 146 .9145 .9612 .9153 .9616 .9161 .9620 .9169 .9623 .9177 .9627 .9185 .9631 147 .9193 .9635 .9201 .9638 .9209 .9642 .9217 .9646 .9225 .9650 .9233 .9653 148 .9240 .9657 .9248 .9660 .9256 .9664 .9263 .9668 .9271 .9671 .9278 .9675 149 .9286 .9678 .9293 .9682 .9301 .9685 .9308 .9689 .9316 .9692 .9323 .9695 150 .9330 .9699 .9337 .9702 .9345 .9706 .9352 .9709 .9359 .9712 .9366 .9716 151 .9373 .9719 .9380 .9722 .9387 .9725 .9394 .9729 .9401 .9732 .9408 .9735 152 .9415 .9738 .9422 .9741 .9428 .9744 .9435 .9747 .9442 .9751 .9448 .9754 153 .9455 .9757 .9462 .9760 .9468 .9763 .9475 .9766 .9481 .9769 .9488 .9772 154 .9494 .9774 .9500 .9777 .9507 .9780 .9513 .9783 .9519 .9786 .9525 .9789 155 .9532 .9792 .9538 .9794 .9544 .9797 .9550 .9800 .9556 .9803 .9562 .9805 156 .9568 .9808 .9574 .9811 .9579 .9813 .9585 .9816 .9591 .9819 .9597 .9821 157 .9603 .9824 .9608 .9826 .9614 .9829 .9619 .9831 .9625 .9834 .9630 .9836 158 .9636 .9839 .9641 .9841 .9647 .9844 .9652 .9846 .9657 .9849 .9663 .9851 159 .9668 .9853 .9673 .9856 .9678 .9858 .9683 .9860 .9688 .9863 .9693 .9865 160 .9698 .9867 .9703 .9869 .9708 .9871 .9713 .9874 .9718 .9876 .9723 .9878 161 .9728 .9880 .9732 .9882 .9737 .9884 .9742 .9886 .9746 .9888 .9751 .9890 162 .9755 .9892 .9760 .9894 .9764 .9896 .9769 .9898 .9773 .9900 .9777 .9902 163 .9782 .9904 .9786 .9906 .9790 .9908 .9794 .9910 .9798 .9911 .9802 .9913 164 .9806 .9915 .9810 .9917 .9814 .9919 .9818 .9920 .9822 .9922 .9826 .9923 165 .9830 .9925 .9833 .9927 .9837 .9929 .9841 .9930 .9844 .9932 .9848 .9933 166 .9851 .9935 .9855 .9937 .9858 .9938 .9862 .9940 .9865 .9941 .9869 .9943 167 .9872 .9944 .9875 .9945 .9878 .9947 .9881 .9948 .9885 .9950 .9888 .9951 168 .9891 .9952 .9894 .9954 .9897 .9955 .9900 .9956 .9903 .9957 .9905 .9959 169 .9908 .9960 .9911 .9961 .9914 .9962 .9916 .9963 .9919 .9965 .9921 .9966 170 .9924 .9967 .9927 .9968 .9929 .9969 .9931 .9970 .9934 .9971 .9936 .9972 171 .9938 .9973 .9941 .9974 .9943 .9975 .9945 .9976 .9947 .9977 .9949 9978 172 .9951 .9979 .9953 .9980 .9955 .9981 .9957 .9981 .9959 .9982 .9961 .9983 173 .9963 .9984 .9964 .9984 .9966 .9985 .9968 .9986 .9969 .9987 .9971 .9987 174 .9973 .9988 .9974 .9988 .9976 .9989 .9977 .9990 .9978 .9991 .9980 .9991 175 .9981 .9992 .9982 .9992 .9983 .9993 .9985 .9993 .9986 .9994 .9987 .9994 176 .9988 .9995 .9989 .9995 .9990 .9996 .9991 .9996 .9992 .9996 .9992 .9997 177 .9993 .9997 .9994 .9997 .9995 .9998 .9995 .9998 .9996 .9998 .9996 .9998 178 .9997 .9999 .9997 .9999 .9998 .9999 .9998 .9999 .9999 .9999 .9999 .9999 179 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 0.0000 1.0000 .0000 INDEX Abscissa, 6. Absolute value, of a directed quan- tity, 7. Addition, of angles, 9; formulas in trigonometry, 95. Angle, definition of, 7; directed, 7; measurement of, 8 ; addition and subtraction of, 9 ; functions of, 2 ; of elevation and depression, 16; of triangle, 48 ; in artillery service, 76. Annuities, 70. Arc of a circle, 76. Artillery service, use of angles in, 76. Axes, of coordinates, 5. Briggian logarithms, 54. Characteristic of a logarithm, 54. Cologarithms, 59. Common logarithms, 54. Compass, Mariner's, 29. Computation, numerical, logarithmic, 61 ff. Coordinates in a plane, 5. Cosecant, 32. Cosine, definition of, 12 : of, 81 ; graph of, 82 ; 40. Cotangent, definition of, 32 Course, 29. Coversed sine, 32. 18, 24 ; variation law of — s, Dead reckoning, 30. Departure, 29. Difference in latitude, 29 ; in longi- tude, 30. Directed, angles, 7 ; quantities, 6 segments, 7. Distance, 29. Elements of a triangle, 1. Function, definition of, 3 ; representa- tion of, 32 ; trigonometric, 12 ff ., 58. Graph of trigonometric functions, 80, 82, 83. Haversine, definition of, 32; solu- tion of triangles by, 48 ; tables of, 117-9. Identities, trigonometric, 35. Initial position, 7. Interest, 70. Interpolation, 22. Knot, 29. Latitude, difference in, 29 ; middle, 30. Law, of sines, 40 ; cosines, 40 ; of tangents, 47. Logarithm, definition of, 52 ; inven- tion of, 50 ; laws of, 53 ; systems of, 54 ; characteristic and man- tissa of, 54 ; use of tables of, 56 ; tables of, 110-16. Logarithmic scale, 73. Magnitude, 6. Mantissa, 54. Mariner's compass, 29. Middle latitude, 30. Mil, 76. Napier, J., 50. Nautical mile, 29. Navigation, 28 ff . Negative angle, definition of, 7; functions of, 85. Ordinate, 6. 121 122 INDEX Parts of a triangle, 1. Period of trigonometric functions, 80, 82, 84. Plane sailing, 28. Plane trigonometry, 1. Product formulas, 101. Projectile, 72. Projection, 92. Quadrant, 6. Radian, 75. Radius of inscribed circle, 46. Rotation, angles of, 8. Rounded numbers, 25. Scale, logarithmic, 73. Secant, definition of, 32. Significant figures, 25. Sine, definition of, 12 ; variation of, 79 ; graph of, 80 ; law of s, 40. Slide rule, 74. Solution of triangles, 1, 16 ff., 41 ft*., 48, 62 ff. Spherical trigonometry, 1. Tables, of squares, 27, 106-7; of haversines, 117-9; of logarithms, 110-11 ; of trigonometric func- tions, 112-19. Tangent, definition of, 3, 12 ; variation of, 82 ; graph of, 83 ; line repre- sentation of, 83 ; law of s, 47. Triangle, area of, 45 ; angles of, 48 ; solution of, 1, 16 ff., 41 ff., 48, 62. Trigonometric equations, 88. Trigonometric functions, definitions of, 3, 12, 15, 32 ; graphs of, 80, 82, 83 ; computation of, 18 ff . ; periods of, 80, 82, 84; inverse, 87; formulas, 15, 32, 34, 96 ff. ; logarithms of, 61 ; tables of, 21, 112-19. Versed sine, defined, 32. Printed in the United States of America. \}) ^ -. At- V (.^ -] ^ <Un> 3-f- - Aw h i~ \ V w*. . ^ a-*, ^ £ 6 f^U^-W^J T^HE following pages contain advertisements of a few of the Macmillan books on kindred subjects. ELEMENTARY MATHEMATICAL ANALYSIS BY JOHN WESLEY YOUNG Professor of Mathematics in Dartmouth College And FRANK MILLET MORGAN Assistant Professor of Mathematics in Dartmouth College Edited by Earle Raymond Hedrick, Professor of Mathematics in the University of Missouri 77/., Cloth, i2tno, $2.60 1 . A textbook for the freshman year in colleges, universities, and technical schools, giving a unified treatment of the essentials of trigonometry, college algebra, and analytic geometry, and intro- ducing the student to the fundamental conceptions of calculus. The various subjects are unified by the great centralizing theme of functionality so that each subject, without losing its fundamental character, is shown clearly in its relationship to the others, and to mathematics as a whole. More emphasis is placed on insight and understanding of fundamental conceptions and modes of thought ; less emphasis on algebraic technique and facility of manipulation. Due recog- nition is given to the cultural motive for the study of mathe- matics and to the disciplinary value. The text presupposes only the usual entrance requirements in elementary algebra and plane geometry. THE MACMILLAN COMPANY Publishers 64-66 Fifth Avenue New York Trigonometry By ALFRED MONROE KENYON Professor of Mathematics, Purdue University and LOUIS INGOLD Assistant Professor of Mathematics, the University of Missouri Edited by Earle Raymond Hedrick With Brief Tables, 8vo, $1.20 With Complete Tables, 8vo, $1.50 The book contains a minimum of purely theoretical matter. Its entire organization is intended to give a clear view of the meaning and the immediate usefulness of Trigonometry. The proofs, however, are in a form that will not require essential revision in the courses that follow. . . . The number of exercises is very large, and the traditional monotony is broken by illus- trations from a variety of topics. Here, as well as in the text, the attempt is often made to lead the student to think for himself by giving suggestions rather than completed solutions or demonstrations. The text proper is short; what is there gained in space is used to make the tables very complete and usable. Attention is called particularly to the complete and handily arranged table of squares, square roots, cubes, etc. ; by its use the Pythagorean theorem and the Cosine Law become practicable for actual computation. The use of the slide rule and of four-place tables is encouraged for problems that do not demand extreme accuracy. Analytic Geometry and Principles of Algebra By ALEXANDER ZIWET Professor of Mathematics, the University of Michigan and LOUIS ALLEN HOPKINS Instructor in Mathematics, the University of Michigan Edited by Earle Raymond Hedrick Cloth, i2tno, $1.75 This work combines with analytic geometry a number of topics traditionally treated in college algebra that depend upon or are closely associated with geometric sensation. Through this combination it becomes possible to show the student more directly the meaning and the usefulness of these subjects. The treatment of solid analytic geometry follows the more usual lines. But, in view of the application to mechanics, the idea of the vector is given some prominence; and the represen- tation of a function of two variables by contour lines as well as by a surface in space is ex- plained and illustrated by practical examples. The exercises have been selected with great care in order not only to furnish sufficient material for practice in algebraic work but also to stimulate independent thinking and to point out the applications of the theory to concrete problems. The number of exercises is sufficient to allow the instructor to make a choice. THE MACMILLAN COMPANY Publishers 64-66 Fifth Avenue New York A Short Course in Mathematics By R. E. MORITZ Professor of Mathematics, University of Washington Cloth, i2tno A text containing the material essential for a short course in Freshman Mathematics which is complete in itself, and which contains no more material than the average Freshman can assimilate. The book, will constitute an adequate preparation for further study, and will enable the student to take up the usual course in analytical geometry without any handicap. Among the subjects treated are : Factoring, Radicals, Fractional and Negative Exponents, Imaginary Quantities, Linear and Quadratic Equations ; Coordinates, Simple and Straight Line Graphs, Curve Plotting, Maxima and Minima, Areas; The General Angle and Its Measures, The Trigonometric or Circular Functions, Functions of an Acute Angle; Solution of Right and Oblique Triangles; Exponents and Logarithms; Application of Logarithms to Numerical Exercises, to Mensuration of Plane Figures, and to Mensuration of Solids; The Four Cases of Oblique Triangles, Miscellaneous Problems Involving Triangles. Plane and Spherical Trigonometry By LEONARD M. PASSANO Associate Professor of Mathematics in the Massachusetts Institute of Technology Cloth, 8vo, $1.25 The chief aims of this text are brevity, clarity, and simplicity. The author presents the whole field of Trigonometry in such a way as to make it interesting to students approaching some maturity, and so as to connect the subject with the mathematics the student has pre- viously studied and with that which may follow. CONTENTS PLANE TRIGONOMETRY chapter 6. The Solution of General Triangles . . The Solution of Trigonometric Equa- CHAPTER 1. The Trigonometric Functions of Any tions Angle and Identical Relations among Them 2. Identical Relations Among the Func- SPHERICAL TRIGONOMETRY tions of Related Angles: The Values of the Functions of Certain Angles 8. Fundamental Relations 3. The Solution of Right Triangles. 9. The Solution of Right Spherical Tri Logarithms and Computation by angles Means of Logarithms 10. The Solution of Oblique Spherical 4. Fundamental Identities Triangles 5. The Circular or Radian Measure of an n. The Earth as a Sphere ... Angle. Inverse Trigonometric Func- Answers tions THE MACMILLAN COMPANY Publishers 64-66 Fifth Avenue New York Differential and Integral Calculus By CLYDE E. LOVE, Ph.D. Assistant Professor of Mathematics in the University of Michigan Crown 8vo, $2.10 Presents a first course in the calculus — substantially as the author has taught it at the University of Michigan for a number of years. The follow- ing points may be mentioned as more or less prominent features of the book : In the treatment of each topic the author has presented his material in such a way that he focuses the student's attention upon the fundamental principle involved, insuring his clear understanding of that, and preventing him from being confused by the discussion of a multitude of details. His constant aim has been to prevent the work from degenerating into mere mechanical routine; thus, wherever possible, except in the purely formal parts of the course, he has avoided the summarizing of the theory into rules or formulas which can be applied blindly. The Calculus By ELLERY WILLIAMS DAVIS Professor of Mathematics, the University of Nebraska Assisted by William Charles Brenke, Associate Professor of Mathe- matics, the University of Nebraska Edited by Earle Raymond Hedrick Cloth, semi- flexible, with Tables, i2tno, $2.10 Edition De Luxe, flexible leather binding, $2.50 This book presents as many and as varied applications of the Calculus as it is possible to do without venturing into technical fields whose subject matter is itself unknown and incomprehensible to the student, and without abandoning an orderly presentation of fundamental principles. The same general tendency has led to the treatment of topics with a view toward bringing out their essential usefulness. Rigorous forms of demon- stration are not insisted upon, especially where the precisely rigorous proofs would be beyond the present grasp of the student. Rather the stress is laid upon the student's certain comprehension of that which is done, and his con- viction that the results obtained are both reasonable and useful. At the same time, an effort has been made to avoid those grosser errors and actual misstatements of fact which have often offended the teacher in texts other- wise attractive and teachable. THE MACMILLAN COMPANY Publishers 64-66 Fifth Avenue New Tork S~- f ( l+L-)^ ^-Cv, '• """"■.SSSHSy-."™ 10w-7. Iir,i YB Y/.o9(.1 889757(^^533 THE UNIVERSITY OF CALIFORNIA LIBRARY (,.o 1 rc